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Class 12 Maths Chapter 7 Integrals NCERT Solutions

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Class 12 Maths Chapter 7 Integrals NCERT Solutions

Integrals Ex 7.1 Class 12

Ex 7.1 Class 12 Maths Question 1.
sin 2x
Solution:

Ex 7.1 Class 12 Maths Question 2.
cos 3x
Solution:

Ex 7.1 Class 12 Maths Question 3.
${ e }^{ 2x }$
Solution:

Ex 7.1 Class 12 Maths Question 4.
(ax + c)²
Solution:

Ex 7.1 Class 12 Maths Question 5.
${ sin\quad 2x-4e }^{ 3x }$
Ex 7.1 Class 12 Maths Solution:

Find the following integrals in Exercises 6 to 20 :

Ex 7.1 Class 12 Maths Question 6.
$\int { \left( { 4e }^{ 3x }+1 \right) dx }$
Solution:

Ex 7.1 Class 12 Maths Question 7.
$\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx }$
Solution:

Ex 7.1 Class 12 Maths Question 8.
$\int { { (ax }^{ 2 }+bx+c)dx }$
Solution:

Ex 7.1 Class 12 Maths Question 9.
$\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx }$
Solution:

Ex 7.1 Class 12 Maths Question 10.
$\int { { \left[ \sqrt { x } -\frac { 1 }{ \sqrt { x } } \right] }^{ 2 }dx }$
Solution:

Ex 7.1 Class 12 Maths Question 11.
$\int { \frac { { x }^{ 3 }+{ 5x }^{ 2 }-4 }{ { x }^{ 2 } } dx }$
Solution:

Ex 7.1 Class 12 Maths Question 12.
$\int { \frac { { x }^{ 3 }+3x+4 }{ \sqrt { x } } dx }$
Solution:

Ex 7.1 Class 12 Maths Question 13.
$\int { \frac { { x }^{ 3 }-{ x }^{ 2 }+x-1 }{ x-1 } dx }$
Solution:

Ex 7.1 Class 12 Maths Question 14.
$\int { \left( 1-x \right) \sqrt { x } dx }$
Solution:

Ex 7.1 Class 12 Maths Question 15.
$\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx }$
Solution:

Ex 7.1 Class 12 Maths Question 16.
$\int { (2x-3cosx+{ e }^{ x })dx }$
Solution:

Ex 7.1 Class 12 Maths Question 17.
$\int { \left( { 2x }^{ 2 }-3sinx+5\sqrt { x } \right) dx }$
Solution:

Ex 7.1 Class 12 Maths Question 18.
$\int { secx(secx+tanx)dx }$
Solution:

Ex 7.1 Class 12 Maths Question 19.
$\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } dx }$
Solution:

Ex 7.1 Class 12 Maths Question 20.
$\int { \frac { 2-3sinx }{ { cos }^{ 2 }x } dx }$
Solution:

Choose the correct answer in Exercises 21 and 22.
Ex 7.1 Class 12 Maths Question 21.
The antiderivative $\left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right)$ equals
(a) $\frac { 1 }{ 3 } { x }^{ \frac { 1 }{ 3 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c$
(b) $\frac { 2 }{ 3 } { x }^{ \frac { 2 }{ 3 } }+{ \frac { 1 }{ 2 } x }^{ 2 }+c$
(c) $\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c$
(d) $\frac { 3 }{ 2 } { x }^{ \frac { 3 }{ 2 } }+\frac { 1 }{ 2 } { x }^{ \frac { 1 }{ 2 } }+c$
Solution:

Ex 7.1 Class 12 Maths Question 22.
If $\frac { d }{ dx } f(x)={ 4x }^{ 3 }-\frac { 3 }{ { x }^{ 4 } }$ such that f(2)=0 then f(x) is
(a) ${ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } -\frac { 129 }{ 8 }$

(b) ${ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } +\frac { 129 }{ 8 }$
(c) ${ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } +\frac { 129 }{ 8 }$
(d) ${ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } -\frac { 129 }{ 8 }$
Solution:

Move to Top → Class 12 Integrals Chapter 7

Integrals Ex 7.2 Class 12

Integrate the functions in Exercises 1 to 37:

Ex 7.2 Class 12 Maths Question 1.
$\frac { 2x }{ 1+{ x }^{ 2 } }$
Solution:

Ex 7.2 Class 12 Maths Question 2.
$\frac { { \left( logx \right) }^{ 2 } }{ x }$
Solution:

Ex 7.2 Class 12 Maths Question 3.
$\frac { 1 }{ x+xlogx }$
Solution:

Ex 7.2 Class 12 Maths Question 4.
sinx sin(cosx)
Solution:

Ex 7.2 Class 12 Maths Question 5.
sin(ax+b) cos(ax+b)
Solution:

Ex 7.2 Class 12 Maths Question 6.
$\sqrt { ax+b }$
Solution:

Ex 7.2 Class 12 Maths Question 7.
$x\sqrt { x+2 }$
Solution:

Ex 7.2 Class 12 Maths Question 8.
$x\sqrt { 1+{ 2x }^{ 2 } }$
Solution:

Ex 7.2 Class 12 Maths Question 9.
$(4x+2)\sqrt { { x }^{ 2 }+x+1 }$
Solution:

Ex 7.2 Class 12 Maths Question 10.
$\frac { 1 }{ x-\sqrt { x } }$
Solution:

Ex 7.2 Class 12 Maths Question 11.
$\frac { x }{ \sqrt { x+4 } } ,x>0$
Solution:

Ex 7.2 Class 12 Maths Question 12.
${ { (x }^{ 3 }-1) }^{ \frac { 1 }{ 3 } }.{ x }^{ 5 }$
Solution:

Ex 7.2 Class 12 Maths Question 13.
$\frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } }$
Solution:

Ex 7.2 Class 12 Maths Question 14.
$\frac { 1 }{ x(logx)^{ m } } ,x>0$
Solution:

Ex 7.2 Class 12 Maths Question 15.
$\frac { x }{ 9-4{ x }^{ 2 } }$
Solution:

Ex 7.2 Class 12 Maths Question 16.
${ e }^{ 2x+3 }$
Solution:

Ex 7.2 Class 12 Maths Question 17.
$\frac { x }{ { e }^{ { x }^{ 2 } } }$
Solution:

Ex 7.2 Class 12 Maths Question 18.
$\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } }$
Solution:

Ex 7.2 Class 12 Maths Question 19.
$\frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 }$
Solution:

Ex 7.2 Class 12 Maths Question 20.
$\frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } }$
Solution:

Ex 7.2 Class 12 Maths Question 21.
tan²(2x-3)
Solution:
∫tan²(2x-3)dx = ∫[sec²(2x-3)-1]dx = I
put 2x-3 = t
so that 2dx = dt
I = $\frac { 1 }{ 2 }$ ∫sec²t dt-x+c
= $\frac { 1 }{ 2 }t-x+c$
= $\frac { 1 }{ 2 }tan(2x-3)-x+c$
Ex 7.2 Class 12 Maths Question 22.
sec²(7-4x)
Solution:

Ex 7.2 Class 12 Maths Question 23.
$\frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } }$
Solution:

Ex 7.2 Class 12 Maths Question 24.
$\frac { 2cosx-3sinx }{ 6cosx+4sinx }$
Solution:

Ex 7.2 Class 12 Maths Question 25.
$\frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } }$
Solution:

Ex 7.2 Class 12 Maths Question 26.
$\frac { cos\sqrt { x } }{ \sqrt { x } }$
Solution:

Ex 7.2 Class 12 Maths Question 27.
$\sqrt { sin2x } cos2x$
Solution:

Ex 7.2 Class 12 Maths Question 28.
$\frac { cosx }{ \sqrt { 1+sinx } }$
Solution:

Ex 7.2 Class 12 Maths Question 29.
cotx log sinx
Solution:

Ex 7.2 Class 12 Maths Question 30.
$\frac { sinx }{ 1+cosx }$
Solution:

Ex 7.2 Class 12 Maths Question 31.
$\frac { sinx }{ { (1+cosx) }^{ 2 } }$
Solution:

Ex 7.2 Class 12 Maths Question 32.
$\frac { 1 }{ 1+cotx }$
Solution:

Ex 7.2 Class 12 Maths Question 33.
$\frac { 1 }{ 1-tanx }$
Solution:

Ex 7.2 Class 12 Maths Question 34.
$\frac { \sqrt { tanx } }{ sinxcosx }$
Solution:

Ex 7.2 Class 12 Maths Question 35.
$\frac { { (1+logx) }^{ 2 } }{ x }$
Solution:

Ex 7.2 Class 12 Maths Question 36.
$\frac { (x+1){ (x+logx) }^{ 2 } }{ x }$
Solution:

Ex 7.2 Class 12 Maths Question 37.
$\frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } dx$
Solution:

Choose the correct answer in exercises 38 and 39

Ex 7.2 Class 12 Maths Question 38.
$\int { \frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx }$
(a) 10x – x10 + C
(b) 10x + x10 + C
(c) (10x – x10) + C
(d) log (10x + x10) + C
Solution:

Ex 7.2 Class 12 Maths Question 39.
$\int { \frac { dx }{ { sin }^{ 2 }x{ \quad cos }^{ 2 }x } = }$
(a) tanx + cotx + c
(b) tanx – cotx + c
(c) tanx cotx + c
(d) tanx – cot2x + c
Solution:

Move to Top → Class 12 Integrals Chapter 7

Integrals Ex 7.3 Class 12

Find the integrals of the functions in Exercises 1 to 22.
Ex 7.3 Class 12 Maths Question 1.
sin²(2x+5)
Solution:

Ex 7.3 Class 12 Maths Question 2.
sin3x cos4x
Solution:

Ex 7.3 Class 12 Maths Question 3.
∫cos2x cos4x cos6x dx
Solution:

Ex 7.3 Class 12 Maths Question 4.
∫sin3(2x+1)dx
Solution:

Ex 7.3 Class 12 Maths Question 5.
sin3x cos3x
Solution:

Ex 7.3 Class 12 Maths Question 6.
sinx sin2x sin3x
Solution:

Ex 7.3 Class 12 Maths Question 7.
sin 4x sin 8x
Solution:

Ex 7.3 Class 12 Maths Question 8.
$\frac { 1-cosx }{ 1+cosx }$
Solution:

Ex 7.3 Class 12 Maths Question 9.
$\frac { cosx }{ 1+cosx }$
Solution:

Ex 7.3 Class 12 Maths Question 10.
∫sinx4 dx
Solution:

Ex 7.3 Class 12 Maths Question 11.
cos4 2x
Solution:

Ex 7.3 Class 12 Maths Question 12.
$\frac { { sin }^{ 2 }x }{ 1+cosx }$
Solution:

Ex 7.3 Class 12 Maths Question 13.
$\frac { cos2x-cos2\alpha }{ cosx-cos\alpha }$
Solution:

Ex 7.3 Class 12 Maths Question 14.
$\frac { cosx-sinx }{ 1+sin2x }$
Solution:

Ex 7.3 Class 12 Maths Question 15.
$\int { { tan }^{ 3 }2x\quad sec2x\quad dx=I }$
Solution:

Ex 7.3 Class 12 Maths Question 16.
tan4x
Solution:

Ex 7.3 Class 12 Maths Question 17.
$\frac { { sin }^{ 3 }x+{ cos }^{ 3 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x }$
Solution:

Ex 7.3 Class 12 Maths Question 18.
$\frac { cos2x+{ 2sin }^{ 2 }x }{ { cos }^{ 2 }x }$
Solution:

Ex 7.3 Class 12 Maths Question 19.
$\frac { 1 }{ sinx{ cos }^{ 3 }x }$
Solution:

Ex 7.3 Class 12 Maths Question 20.
$\frac { cos2x }{ { (cosx+sinx) }^{ 2 } }$
Solution:

Ex 7.3 Class 12 Maths Question 21.
sin-1 (cos x)
Solution:

Ex 7.3 Class 12 Maths Question 22.
$\int { \frac { 1 }{ cos(x-a)cos(x-b) } dx }$
Solution:

Ex 7.3 Class 12 Maths Question 23.
$\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } dx\quad is\quad equal\quad to$
(a) tanx+cotx+c
(b) tanx+cosecx+c
(c) -tanx+cotx+c
(d) tanx+secx+c
Solution:

Ex 7.3 Class 12 Maths Question 24.
$\int { \frac { e^{ x }(1+x) }{ cos^{ 2 }({ e }^{ x }.{ x }) } } dx\quad is\quad equal\quad to$
(a) -cot(e.xx)+c
(b) tan(xex)+c
(c) tan(ex)+c
(d) cot ex+c
Solution:

Move to Top → Class 12 Integrals Chapter 7

Integrals Class 12 Ex 7.4

Ex 7.4 Class 12 Maths Question 1.
$\frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 }$
Solution:

Ex 7.4 Class 12 Maths Question 2.
$\frac { 1 }{ \sqrt { 1+{ 4x }^{ 2 } } }$
Solution:

Ex 7.4 Class 12 Maths Question 3.
$\frac { 1 }{ \sqrt { { (2-x) }^{ 2 }+1 } }$
Solution:

Ex 7.4 Class 12 Maths Question 4.
$\frac { 1 }{ \sqrt { 9-{ 25x }^{ 2 } } }$
Solution:

Ex 7.4 Class 12 Maths Question 5.
$\frac { 3x }{ 1+{ 2x }^{ 4 } }$
Solution:

Ex 7.4 Class 12 Maths Question 6.
$\frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } }$
Solution:

Ex 7.4 Class 12 Maths Question 7.
$\frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } }$
Solution:

Ex 7.4 Class 12 Maths Question 8.
$\frac { { x }^{ 2 } }{ \sqrt { { x }^{ 6 }+{ a }^{ 6 } } }$
Solution:

Ex 7.4 Class 12 Maths Question 9.
$\frac { { sec }^{ 2 }x }{ \sqrt { { tan }^{ 2 }x+4 } }$
Solution:

Ex 7.4 Class 12 Maths Question 10.
$\frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } }$
Solution:

Ex 7.4 Class 12 Maths Question 11.

$\frac { 1 }{ { 9x }^{ 2 }+6x+5 }$
Solution:

Ex 7.4 Class 12 Maths Question 12.
$\frac { 1 }{ \sqrt { 7-6x-{ x }^{ 2 } } }$
Solution:

Ex 7.4 Class 12 Maths Question 13.
$\frac { 1 }{ \sqrt { (x-1)(x-2) } }$
Solution:

Ex 7.4 Class 12 Maths Question 14.
$\frac { 1 }{ \sqrt { 8+3x-{ x }^{ 2 } } }$
Solution:

Ex 7.4 Class 12 Maths Question 15.
$\frac { 1 }{ \sqrt { (x-a)(x-b) } }$
Solution:

Ex 7.4 Class 12 Maths Question 16.
$\frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } }$
Solution:

Ex 7.4 Class 12 Maths Question 17.
$\frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } }$
Solution:

Ex 7.4 Class 12 Maths Question 18.
$\frac { 5x-2 }{ 1+2x+{ 3x }^{ 2 } }$
Solution:
put 5x-2=A $\frac { d }{ dx }$ (1+2x+3x²)+B
⇒ 6A=5, A= $\frac { 5 }{ 6 }-2=2A+B$ , B= $-\frac { 11 }{ 3 }$

Ex 7.4 Class 12 Maths Question 19.
$\frac { 6x+7 }{ \sqrt { (x-5)(x-4) } }$
Solution:

Ex 7.4 Class 12 Maths Question 20.
$\frac { x+2 }{ \sqrt { 4x-{ x }^{ 2 } } }$
Solution:

Ex 7.4 Class 12 Maths Question 21.
$\frac { x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } }$
Solution:

Ex 7.4 Class 12 Maths Question 22.
$\frac { x+3 }{ { x }^{ 2 }-2x-5 }$
Solution:

Ex 7.4 Class 12 Maths Question 23.
$\frac { 5x+3 }{ \sqrt { { x }^{ 2 }+4x+10 } }$
Solution:

Ex 7.4 Class 12 Maths Question 24.
$\int { \frac { dx }{ { x }^{ 2 }+2x+2 } equals }$
(a) xtan-1(x+1)+c
(b) (x+1)tan-1x+c
(c) tan-1(x+1)+c
(d) tan-1x+c
Solution:

Ex 7.4 Class 12 Maths Question 25.
$\int { \frac { dx }{ \sqrt { 9x-{ 4x }^{ 2 } } } equals }$
(a) $\frac { 1 }{ 9 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c$
(b) $\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 8x-9 }{ 9 } \right) +c$
(c) $\frac { 1 }{ 3 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c$
(d) ${ sin }^{ -1 }\left( \frac { 9x-8 }{ 9 } \right) +c$
Solution:

Move to Top → Class 12 Integrals Chapter 7

Integrals Class 12 Ex 7.5

Ex 7.5 Class 12 Maths Question 1.
$\frac { x }{ (x+1)(x+2) }$
Solution:

Ex 7.5 Class 12 Maths Question 2.
$\frac { 1 }{ { x }^{ 2 }-9 }$
Solution:

Ex 7.5 Class 12 Maths Question 3.
$\frac { 3x-1 }{ (x-1)(x-2)(x-3) }$
Solution:

Ex 7.5 Class 12 Maths Question 4.
$\frac { x }{ (x-1)(x-2)(x-3) }$
Solution:

Ex 7.5 Class 12 Maths Question 5.
$\frac { 2x }{ { x }^{ 2 }+3x+2 }$
Solution:

Ex 7.5 Class 12 Maths Question 6.
$\frac { 1-{ x }^{ 2 } }{ x(1-2x) }$
Solution:

Ex 7.5 Class 12 Maths Question 7.
$\frac { x }{ \left( { x }^{ 2 }+1 \right) \left( x-1 \right) }$
Solution:

Ex 7.5 Class 12 Maths Question 8.
$\frac { x }{ { \left( x-1 \right) }^{ 2 }\left( x+2 \right) }$
Solution:

Ex 7.5 Class 12 Maths Question 9.
$\frac { 3x+5 }{ { x }^{ 3 }-{ x }^{ 2 }-x+1 }$
Solution:

Ex 7.5 Class 12 Maths Question 10.
$\frac { 2x-3 }{ ({ x }^{ 2 }-1)(2x+3) }$
Solution:

Ex 7.5 Class 12 Maths Question 11.
$\frac { 5x }{ (x-1)({ x }^{ 2 }-4) }$
Solution:

Ex 7.5 Class 12 Maths Question 12.
$\frac { { x }^{ 3 }+x+1 }{ { x }^{ 2 }-1 }$
Solution:

Ex 7.5 Class 12 Maths Question 13.
$\frac { 2 }{ (1-x)(1+{ x }^{ 2 }) }$
Solution:

Ex 7.5 Class 12 Maths Question 14.
$\frac { 3x-1 }{ { (x+2) }^{ 2 } }$
Solution:

Ex 7.5 Class 12 Maths Question 15.
$\frac { 1 }{ { x }^{ 4 }-1 }$
Solution:

Ex 7.5 Class 12 Maths Question 16.
$\frac { 1 }{ x({ x }^{ n }+1) }$
[Hint : multiply numerator and denominator by xn-1 and put xn = t ]
Solution:

Ex 7.5 Class 12 Maths Question 17.
$\frac { cosx }{ (1-sinx)(2-sinx) }$
Solution:

Ex 7.5 Class 12 Maths Question 18.
$\frac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right) }$
Solution:

Ex 7.5 Class 12 Maths Question 19.
$\frac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) }$
Solution:

Ex 7.5 Class 12 Maths Question 20.
$\frac { 1 }{ x({ x }^{ 4 }-1) }$
Solution:

Ex 7.5 Class 12 Maths Question 21.
$\frac { 1 }{ { e }^{ x }-1 }$
Solution:

Ex 7.5 Class 12 Maths Question 22.
choose the correct answer in each of the following :
$\int { \frac { xdx }{ (x-1)(x-2) } equals }$
(a) $log\left| \frac { { (x-1) }^{ 2 } }{ x-2 } \right| +c$
(b) $log\left| \frac { { (x-2) }^{ 2 } }{ x-1 } \right| +c$
(c) $log\left| \left( \frac { x-{ 1 }^{ 2 } }{ x-2 } \right) \right| +c$
(d) log|(x-1)(x-2)|+c
Solution:

Ex 7.5 Class 12 Maths Question 23.
$\int { \frac { dx }{ x({ x }^{ 2 }+1) } equals }$
(a) $log|x|-\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c$
(b) $log|x|+\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c$
(c) $-log|x|+\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c$
(d) $\frac { 1 }{ 2 } log|x|+log({ x }^{ 2 }+1)+c$
Solution:

Move to Top → Class 12 Integrals Chapter 7

Integrals Class 12 Ex 7.6

Ex 7.6 Class 12 Maths Question 1.
x sinx
Solution:

Ex 7.6 Class 12 Maths Question 2.
x sin3x
Solution:

Ex 7.6 Class 12 Maths Question 3.
${ x }^{ 2 }{ e }^{ x }$
Solution:

Ex 7.6 Class 12 Maths Question 4.
x logx
Solution:

Ex 7.6 Class 12 Maths Question 5.
x log2x
Solution:

Ex 7.6 Class 12 Maths Question 6.
${ x }^{ 2 }logx$
Solution:

Ex 7.6 Class 12 Maths Question 7.
$x\quad { sin }^{ -1 }x$
Solution:

Ex 7.6 Class 12 Maths Question 8.
$x\quad { tan }^{ -1 }x$
Solution:

Ex 7.6 Class 12 Maths Question 9.
$x\quad { cos }^{ -1 }x$
Solution:

Ex 7.6 Class 12 Maths Question 10.
${ (sin }^{ -1 }{ x })^{ 2 }$
Solution:

Ex 7.6 Class 12 Maths Question 11.
$\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } }$
Solution:

Ex 7.6 Class 12 Maths Question 12.
x sec²x
Solution:

Ex 7.6 Class 12 Maths Question 13.
${ ta }n^{ -1 }x$
Solution:

Ex 7.6 Class 12 Maths Question 14.
x(logx)²
Solution:

Ex 7.6 Class 12 Maths Question 15.
(x²+1)logx
Solution:

Ex 7.6 Class 12 Maths Question 16.
${ e }^{ x }(sinx+cosx)$
Solution:

Ex 7.6 Class 12 Maths Question 17.
$\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } }$
Solution:

Ex 7.6 Class 12 Maths Question 18.
$\frac { { e }^{ x }(1+sinx) }{ 1+cosx }$
Solution:

Ex 7.6 Class 12 Maths Question 19.
${ e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right)$
Solution:

Ex 7.6 Class 12 Maths Question 20.
$\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } }$
Solution:

Ex 7.6 Class 12 Maths Question 21.
${ e }^{ 2x }sinx$
Solution:

Ex 7.6 Class 12 Maths Question 22.
${ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right)$
Solution:

Choose the correct answer in exercise 23 and 24

Ex 7.6 Class 12 Maths Question 23.
$\int { { x }^{ 2 }{ e }^{ { x }^{ 3 } } } dx\quad equals$
(a) $\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c$
(b) $\frac { 1 }{ 3 } +{ e }^{ { x }^{ 2 } }+c$
(c) $\frac { 1 }{ 2 } { e }^{ { x }^{ 3 } }+c$
(d) $\frac { 1 }{ 2 } { e }^{ { x }^{ 2 } }+c$
Solution:

Ex 7.6 Class 12 Maths Question 24.
$\int { { e }^{ x }secx(1+tanx) } dx\quad equals$
(a) ${ e }^{ x }cosx+c$
(b) ${ e }^{ x }secx+c$
(c) ${ e }^{ x }sinx+c$
(d) ${ e }^{ x }tanx+c$
Solution:

Move to Top → Class 12 Integrals Chapter 7

Class 12 Integrals Ex 7.7

Integral the function in exercises 1 to 9

Ex 7.7 Class 12 Maths Question 1.
$\sqrt { 4-{ x }^{ 2 } }$
Solution:

Ex 7.7 Class 12 Maths Question 2.
$\sqrt { 1-{ 4x }^{ 2 } }$
Solution:

Ex 7.7 Class 12 Maths Question 3.
$\sqrt { { x }^{ 2 }+4x+6 }$
Solution:

Ex 7.7 Class 12 Maths Question 4.
$\sqrt { { x }^{ 2 }+4x+1 }$
Solution:

Ex 7.7 Class 12 Maths Question 5.
$\sqrt { 1-4x-{ x }^{ 2 } }$
Solution:

Ex 7.7 Class 12 Maths Question 6.
$\sqrt { { x }^{ 2 }+4x-5 }$
Solution:

Ex 7.7 Class 12 Maths Question 7.
$\sqrt { 1+3x-{ x }^{ 2 } }$
Solution:

Ex 7.7 Class 12 Maths Question 8.
$\sqrt { { x }^{ 2 }+3x }$
Solution:

Ex 7.7 Class 12 Maths Question 9.
$\sqrt { 1+\frac { { x }^{ 2 } }{ 9 } }$
Solution:

Choose the correct answer in the Exercises 10 to 11:

Ex 7.7 Class 12 Maths Question 10.
$\int { \sqrt { 1+{ x }^{ 2 } } } dx\quad is\quad equal\quad to$
(a) $\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log|x+\sqrt { 1+{ x }^{ 2 } } |+c$
(b) $\frac { 2 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c$
(c) $\frac { 2 }{ 3 } x{ \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c$
(d) $\frac { { x }^{ 2 } }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } { x }^{ 2 }log\left| x+\sqrt { 1+{ x }^{ 2 } } \right| +c$
Solution:

Ex 7.7 Class 12 Maths Question 11.
$\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\quad is\quad equal\quad to$

Solution:

Move to Top → Class 12 Integrals Chapter 7

Class 12 Integrals Ex 7.8

Ex 7.8 Class 12 Maths Question 1.
$\int _{ a }^{ b }{ x\quad dx }$
Solution:

Ex 7.8 Class 12 Maths Question 2.
$\int _{ 0 }^{ 5 }{ (x+1)dx }$
Solution:

Ex 7.8 Class 12 Maths Question 3.
$\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx$
Solution:

Ex 7.8 Class 12 Maths Question 4.
$\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx$
Solution:

Ex 7.8 Class 12 Maths Question 5.
$\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad$
Solution:

Ex 7.8 Class 12 Maths Question 6.
$\int _{ 0 }^{ 4 }{ { (x+e }^{ 2x }) } dx\quad$
Solution:

Move to Top → Class 12 Integrals Chapter 7

Class 12 Integrals Ex 7.9

Ex 7.9 Class 12 Maths Question 1.
$\int _{ -1 }^{ 1 }{ { (x+1 }) } dx\quad$
Solution:

Ex 7.9 Class 12 Maths Question 2.
$\int _{ 2 }^{ 3 }{ \frac { 1 }{ x } dx }$
Solution:

Ex 7.9 Class 12 Maths Question 3.
$\int _{ 1 }^{ 2 }{ \left( { 4x }^{ 3 }-{ 5x }^{ 2 }+6x+9 \right) dx }$
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Ex 7.9 Class 12 Maths Question 4.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin2x\quad dx }$
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Ex 7.9 Class 12 Maths Question 5.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cos2x\quad dx }$
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Ex 7.9 Class 12 Maths Question 6.
$\int _{ 4 }^{ 5 }{ { e }^{ x }dx }$
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Ex 7.9 Class 12 Maths Question 7.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ tanx\quad dx }$
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Ex 7.9 Class 12 Maths Question 8.
$\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }{ cosec\quad xdx }$
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Ex 7.9 Class 12 Maths Question 9.
$\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 1-{ x }^{ 2 } } } }$
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Ex 7.9 Class 12 Maths Question 10.
$\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } }$
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Ex 7.9 Class 12 Maths Question 11.
$\int _{ 2 }^{ 3 }{ \frac { dx }{ { x }^{ 2 }-1 } }$
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Ex 7.9 Class 12 Maths Question 12.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 } } xdx$
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Ex 7.9 Class 12 Maths Question 13.
$\int _{ 2 }^{ 3 }{ \frac { x }{ { x }^{ 2 }+1 } } dx$
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Ex 7.9 Class 12 Maths Question 14.
$\int _{ 0 }^{ 1 }{ \frac { 2x+3 }{ { 5x }^{ 2 }+1 } dx }$
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Ex 7.9 Class 12 Maths Question 15.
$\int _{ 0 }^{ 1 }{ { xe }^{ { x }^{ 2 } }dx }$
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Ex 7.9 Class 12 Maths Question 16.
$\int _{ 1 }^{ 2 }{ \frac { { 5x }^{ 2 } }{ { x }^{ 2 }+4x+3 } dx }$
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Ex 7.9 Class 12 Maths Question 17.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx }$
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Ex 7.9 Class 12 Maths Question 18.
$\int _{ 0 }^{ \pi }{ \left( { sin }^{ 2 }\frac { x }{ 2 } -{ cos }^{ 2 }\frac { x }{ 2 } \right) } dx$
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Ex 7.9 Class 12 Maths Question 19.
$\int _{ 0 }^{ 2 }{ \frac { 6x+3 }{ { x }^{ 2 }+4 } } dx$
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Ex 7.9 Class 12 Maths Question 20.
$\int _{ 0 }^{ 1 }{ \left( { xe }^{ x }+sin\frac { \pi x }{ 4 } \right) dx }$
Solution:

Ex 7.9 Class 12 Maths Question 21.
$\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ { 1+x }^{ 2 } } \quad equals }$
(a) $\frac { \pi }{ 3 }$
(b) $\frac { 2\pi }{ 3 }$
(c) $\frac { \pi }{ 6 }$
(d) $\frac { \pi }{ 12 }$
Solution:

Ex 7.9 Class 12 Maths Question 22.
$\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ 4+{ 9x }^{ 2 } } equals }$
(a) $\frac { \pi }{ 6 }$
(b) $\frac { \pi }{ 12 }$
(c) $\frac { \pi }{ 24 }$
(d) $\frac { \pi }{ 4 }$
Solution:

Move to Top → Class 12 Integrals Chapter 7

Integration Class 12 Ex 7.10

Ex 7.10 Class 12 Maths Question 1:

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Ex 7.10 Class 12 Maths Question 2:

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Ex 7.10 Class 12 Maths Question 9:

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Ex 7.10 Class 12 Maths Question 11:

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Ex 7.10 Class 12 Maths Question 12:

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Ex 7.10 Class 12 Maths Question 13:

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Ex 7.10 Class 12 Maths Question 18:

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Ex 7.10 Class 12 Maths Question 19:

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Ex 7.10 Class 12 Maths Question 20:

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Move to Top → Class 12 Integrals Chapter 7

NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

Move to Top → Class 12 Integrals Chapter 7

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Miscellaneous Exercise Class 11 Maths Question 1:

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Miscellaneous Exercise Class 11 Maths Question 12:

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Miscellaneous Exercise Class 11 Maths Question 13:

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Miscellaneous Exercise Class 11 Maths Question 14:

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Miscellaneous Exercise Class 11 Maths Question 16:

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Miscellaneous Exercise Class 11 Maths Question 17:

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Miscellaneous Exercise Class 11 Maths Question 18:

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Miscellaneous Exercise Class 11 Maths Question 20:

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Miscellaneous Exercise Class 11 Maths Question 21:

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Miscellaneous Exercise Class 11 Maths Question 24:

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Miscellaneous Exercise Class 11 Maths Question 25:

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Miscellaneous Exercise Class 11 Maths Question 43:

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Miscellaneous Exercise Class 11 Maths Question 44:

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Move to Top → Class 12 Integrals Chapter 7

Integration Formulas

Class 12 Maths Chapter 7 Integrals NCERT Solutions

Class 12 Maths Chapter 7 Integrals NCERT Solutions

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