NCERT solutions for Mathematics Part 1 and 2 [English] Class 12 chapter 7 - Integrals [Latest edition]

Find an anti derivative (or integral) of the following function by the method of inspection.

sin 2x

Find an anti derivative (or integral) of the following function by the method of inspection.

Cos 3x

Find an anti derivative (or integral) of the following function by the method of inspection.

e^2x

Find an anti derivative (or integral) of the following function by the method of inspection.

(axe + b)²

Find an antiderivative (or integral) of the following function by the method of inspection.

sin 2x – 4 e^3x

Find the following integrals:

`int (4e^(3x) + 1)`

Find the following integrals:

`intx^2 (1 - 1/x^2)dx`

Find the following integrals:

`int (ax^2 + bx + c) dx`

Find the following integrals:

`int(2x^2 + e^x)dx`

Find the following integrals:

`int(sqrtx - 1/sqrtx)^2 dx`

Find the following integrals:

`int (x^3 + 5x^2   -4)/x^2 dx`

Find the following integrals:

`int (x^3 + 3x + 4)/sqrtx dx`

Find the following integrals:

`int (x^3 - x^2 + x - 1)/(x - 1) dx`

Find the following integrals:

`int(1 - x) sqrtx dx`

Find the following integrals:

`intsqrtx( 3x^2 + 2x + 3) dx`

Find the following integrals:

`int(2x - 3cos x + e^x) dx`

Find the following integrals:

`int(2x^2 - 3sinx + 5sqrtx) dx`

Find the following integrals:

`intsec x (sec x + tan x) dx`

Find the following integrals:

`int(sec^2x)/(cosec^2x) dx`

Find the following integrals:

`int (2 - 3 sinx)/(cos^2 x) dx.`

The anti derivative of `(sqrtx + 1/ sqrtx)` equals:

If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.

Integrate the functions:

`(2x)/(1 + x^2)`

Integrate the functions:

`(log x)^2/x`

Integrate the functions:

`1/(x + x log x)`

Integrate the functions:

sin x ⋅ sin (cos x)

Integrate the functions:

sin (ax + b) cos (ax + b)

Integrate the functions:

`sqrt(ax + b)`

Integrate the functions:

`xsqrt(x + 2)`

Integrate the functions:

`xsqrt(1+ 2x^2)`

Integrate the functions:

(4x + 2) `sqrt(x^2 + x +1)`

Integrate the functions:

`1/(x-sqrtx)`

Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 

Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 

Integrate the functions:

`(x^3 - 1)^(1/3) x^5`

Integrate the functions:

`x^2/(2+ 3x^3)^3`

Integrate the functions:

`1/(x(log x)^m),  x > 0, m ne 1`

Integrate the functions:

`x/(9 - 4x^2)`

Integrate the functions:

`e^(2x+3)`

Integrate the functions:

`x/(e^(x^2))`

Integrate the functions:

`e^(tan^(-1)x)/(1+x^2)`

Integrate the functions:

`(e^(2x) - 1)/(e^(2x) + 1)`

Integrate the functions:

`(e^(2x) -  e^(-2x))/(e^(2x) + e^(-2x))`

Integrate the functions:

tan²(2x – 3)

Integrate the functions:

sec²(7 – 4x)

Integrate the functions:

`(sin^(-1) x)/(sqrt(1-x^2))`

Integrate the functions:

`(2cosx - 3sinx)/(6cos x + 4 sin x)`

Integrate the functions:

`1/(cos^2 x(1-tan x)^2`

Integrate the functions:

`cos sqrt(x)/sqrtx`

Integrate the functions:

`sqrt(sin 2x) cos 2x`

Integrate the functions:

`cos x /(sqrt(1+sinx))`

Integrate the functions:

cot x log sin x

Integrate the functions:

`sin x/(1+ cos x)`

Integrate the functions:

`(sin x)/(1+ cos x)^2`

Integrate the functions:

`1/(1 + cot x)`

Integrate the functions:

`1/(1 - tan x)`

Integrate the functions:

`sqrt(tanx)/(sinxcos x)`

Integrate the functions:

`(1+ log x)^2/x`

Integrate the functions:

`((x+1)(x + logx)^2)/x`

Integrate the functions:

`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`

`(10x^9 + 10^x log_e 10)/(x^10 + 10^x)  dx` equals:

`int (dx)/(sin^2 x cos^2 x)` equals:

Find the integrals of the function:

sin² (2x + 5)

Find the integrals of the function:

sin 3x cos 4x

Find the integrals of the function:

cos 2x cos 4x cos 6x

Find the integrals of the function:

sin³ (2x + 1)

Find the integrals of the function:

sin³ x cos³ x

Find the integrals of the function:

sin x sin 2x sin 3x

Find the integrals of the function:

sin 4x sin 8x

Find the integrals of the function:

`(1-cosx)/(1 +  cos x)`

Find the integrals of the function:

`cos x/(1 + cos x)`

Find the integrals of the function:

sin⁴ x

Find the integrals of the function:

cos⁴ 2x

Find the integrals of the function:

`(sin^2 x)/(1 + cos x)`

Find the integrals of the function:

`(cos 2x - cos 2 alpha)/(cos x - cos alpha)`

Find the integrals of the function:

`(cos x -  sinx)/(1+sin 2x)`

Find the integrals of the function:

tan³ 2x sec 2x

Find the integrals of the function:

tan⁴x

Find the integrals of the function:

`(sin^3 x + cos^3 x)/(sin^2x cos^2 x)`

Find the integrals of the function:

`(cos 2x+ 2sin^2x)/(cos^2 x)`

Find the integrals of the function:

`1/(sin xcos^3 x)`

Find the integrals of the function:

`(cos 2x)/(cos x + sin x)^2`

Find the integrals of the function:

sin⁻¹ (cos x)

Find the integrals of the function:

`1/(cos(x - a) cos(x - b))`

`int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx` is equal to ______.

`int (e^x(1 +x))/cos^2(e^x x) dx` equals ______.

Integrate the function `(3x^2)/(x^6 + 1)`

Integrate the function `1/sqrt(1+4x^2)`

Integrate the function `1/sqrt((2-x)^2 + 1)`

Integrate the function `1/sqrt(9 - 25x^2)`

Integrate the function `(3x)/(1+ 2x^4)`

Integrate the function `x^2/(1 - x^6)`

Integrate the function `(x - 1)/sqrt(x^2 - 1)`

Integrate the function `x^2/sqrt(x^6 + a^6)`

Integrate the function `(sec^2 x)/sqrt(tan^2 x + 4)`

Integrate the function `1/sqrt(x^2 +2x + 2)`

Integrate the function `1/(9x^2 + 6x + 5)`

Integrate the function `1/sqrt(7 - 6x - x^2)`

Integrate the function `1/sqrt((x -1)(x - 2))`

Integrate the function `1/sqrt(8+3x  - x^2)`

Integrate the function `1/sqrt((x - a)(x - b))`

Integrate the function `(4x+ 1)/sqrt(2x^2 + x - 3)`

Integrate the function `(x + 2)/sqrt(x^2 -1)`

Integrate the function `(5x - 2)/(1 + 2x + 3x^2)`

Integrate the function `(6x + 7)/sqrt((x - 5)(x - 4))`

Integrate the function `(x + 2)/sqrt(4x - x^2)`

Integrate the function `(x+2)/sqrt(x^2 + 2x + 3)`

Integrate the function `(x + 3)/(x^2 - 2x - 5)`

Integrate the function `(5x + 3)/sqrt(x^2 + 4x + 10)`

`int dx/(x^2 + 2x + 2)` equals:

`int dx/sqrt(9x - 4x^2)` equals:

Integrate the rational function:

`x/((x + 1)(x+ 2))`

Integrate the rational function:

`1/(x^2 - 9)`

Integrate the rational function:

`(3x - 1)/((x - 1)(x - 2)(x - 3))`

Integrate the rational function:

`x/((x-1)(x- 2)(x - 3))`

Integrate the rational function:

`(2x)/(x^2 + 3x + 2)`

Integrate the rational function:

`(1 - x^2)/(x(1-2x))`

Integrate the rational function:

`x/((x^2+1)(x - 1))`

Integrate the rational function:

`x/((x -1)^2 (x+ 2))`

Integrate the rational function:

`(3x + 5)/(x^3 - x^2 - x + 1)`

Integrate the rational function:

`(2x - 3)/((x^2 -1)(2x + 3))`

Integrate the rational function:

`(5x)/((x + 1)(x^2 - 4))`

Integrate the rational function:

`(x^3 + x + 1)/(x^2 -1)`

Integrate the rational function:

`2/((1-x)(1+x^2))`

Integrate the rational function:

`(3x -1)/(x + 2)^2`

Integrate the rational function:

`1/(x^4 - 1)`

Integrate the rational function:

`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by x^{n − 1} and put xⁿ = t]

Integrate the rational function:

`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]

Integrate the rational function:

`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`

Integrate the rational function:

`(2x)/((x^2 + 1)(x^2 + 3))`

Integrate the rational function:

`1/(x(x^4 - 1))`

Integrate the rational function:

`1/(e^x -1)`[Hint: Put e^x = t]

`int (xdx)/((x - 1)(x - 2))` equals:

`int (dx)/(x(x^2 + 1))` equals:

Integrate the function in x sin x.

Integrate the function in x sin 3x.

Integrate the function in `x^2e^x`.

Integrate the function in x log x.

Integrate the function in x log 2x.

Integrate the function in x² log x.

Integrate the function in x sin⁻¹ x.

Integrate the function in x tan^-1 x.

Integrate the function in x cos^-1 x.

Integrate the function in (sin^-1x)².

Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.

Integrate the function in x sec² x.

Integrate the function in tan^-1 x.

Integrate the function in x (log x)².

Integrate the function in (x² + 1) log x.

Integrate the function in e^x (sinx + cosx).

Integrate the function in `(xe^x)/(1+x)^2`.

Integrate the function in `e^x (1 + sin x)/(1+cos x)`.

Integrate the function in `e^x (1/x - 1/x^2)`.

Integrate the function in `((x- 3)e^x)/(x - 1)^3`.

Integrate the function in e^2x sin x.

Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.

`intx^2 e^(x^3) dx` equals: 

`int e^x sec x (1 +   tan x) dx` equals:

Integrate the function:

`sqrt(4 - x^2)`

Integrate the function:

`sqrt(1- 4x^2)`

Integrate the function:

`sqrt(x^2 + 4x + 6)`

Integrate the function:

`sqrt(x^2 + 4x +1)`

Integrate the function:

`sqrt(1-4x - x^2)`

Integrate the function:

`sqrt(x^2 + 4x - 5)`

Integrate the function:

`sqrt(1+ 3x - x^2)`

Integrate the function:

`sqrt(x^2 + 3x)`

Integrate the function:

`sqrt(1+ x^2/9)`

`int sqrt(1+ x^2)  dx` is equal to ______.

`int sqrt(1+ x^2)  dx` is equal to ______.

`int sqrt(x^2 - 8x + 7) dx` is equal to ______.

Evaluate the following definite integrals as limit of sums.

`int_a^b x dx`

Evaluate the following definite integrals as limit of sums.

`int_0^5 (x+1) dx`

Evaluate the following definite integrals as limit of sums. 

`int_2^3 x^2 dx`

Evaluate the following definite integrals as limit of sums.

`int_1^4 (x^2 - x) dx`

Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`

Evaluate the following definite integrals as limit of sums.

`int_0^4 (x + e^(2x)) dx`

Evaluate the definite integral:

`int_(-1)^1 (x + 1)dx`

Evaluate the definite integral:

`int_2^3 1/x dx`

Evaluate the integral by using substitution.

`int_0^1 sin^(-1) ((2x)/(1+ x^2)) dx`

Evaluate the definite integral:

`int_1^2 (4x^3 - 5x^2 + 6x + 9)  dx`

Evaluate the definite integral:

`int_0^(pi/4) sin2xdx`

Evaluate the definite integral:

`int_0^(pi/2) cos 2x  dx`

Evaluate the definite integral:

`int_4^5 e^x dx`

Evaluate the definite integral:

`int_0^(pi/4) tan x dx`

Evaluate the definite integral:

`int_(pi/6)^(pi/4) cosec x  dx`

Evaluate the definite integral:

`int_0^1 dx/sqrt(1-x^2)`

Evaluate the definite integral:

`int_0^1 dx/(1+x^2)`

If `f(x) = int_0^pi t sin  t  dt`, then f' (x) is ______.

Evaluate the definite integral:

`int_2^3 dx/(x^2 - 1)`

Evaluate the definite integral:

`int_0^(pi/2) cos^2 xdx`

Evaluate the definite integral:

`int_2^3 (xdx)/(x^2 + 1)`

Evaluate the definite integral:

`int_0^1 (2x + 3)/(5x^2 + 1) dx`

Evaluate the definite integral:

`int_0^1 x e^(x^2) dx`

Evaluate the definite integral:

`int_1^2 (5x^2)/(x^2 + 4x + 3)`

Evaluate the definite integral:

`int_0^(pi/4) (2 sec^2 x + x^3  + 2) dx`

Evaluate the definite integral:

`int_0^pi (sin^2  x/2 - cos^2  x/2) dx`

Evaluate the definite integral:

`int_0^2 (6x +3)/(x^2 + 4)` dx

Evaluate the definite integral:

`int_0^1 (xe^x + sin  (pix)/4)`

`int_1^(sqrt3)dx/(1+x^2) ` equals:

`int_0^(2/3) dx/(4+9x^2)` equals:

Evaluate the integral by using substitution.

`int_0^1 x/(x^2 +1)`dx

Evaluate the integral by using substitution.

`int_0^(pi/2) sqrt(sin phi) cos^5 phidphi`

Evaluate the integral by using substitution.

`int_0^2 xsqrt(x+2)`  (Put x + 2 = `t^2`)

Evaluate the integral by using substitution.

`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`

Evaluate the integral by using substitution.

`int_0^2 dx/(x + 4 - x^2)`

Evaluate the integral by using substitution.

`int_(-1)^1 dx/(x^2 + 2x  + 5)`

Evaluate the integral by using substitution.

`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`

The value of the integral `int_(1/3)^4 ((x- x^3)^(1/3))/x^4` dx is ______.

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) cos^2 x dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2)  sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx` 

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2)  (cos^5  xdx)/(sin^5 x + cos^5 x)`

By using the properties of the definite integral, evaluate the integral:

`int_(-5)^5 | x + 2| dx`

By using the properties of the definite integral, evaluate the integral:

`int_2^8 |x - 5| dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^1 x(1-x)^n dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/4) log (1+ tan x) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^2 xsqrt(2 -x)dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) (2log sin x - log sin 2x)dx`

By using the properties of the definite integral, evaluate the integral:

`int_((-pi)/2)^(pi/2) sin^2 x  dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^pi (x  dx)/(1+ sin x)`

By using the properties of the definite integral, evaluate the integral:

`int_(pi/2)^(pi/2) sin^7 x dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(2x) cos^5 xdx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^pi log(1+ cos x) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^a  sqrtx/(sqrtx + sqrt(a-x))   dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^4 |x - 1| dx`

Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx`  if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.

`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.

The value of `int_0^(pi/2) log  ((4+ 3sinx)/(4+3cosx))` dx is ______.

Integrate the function:

`1/(x - x^3)`

Integrate the function:

`1/(sqrt(x+a) + sqrt(x+b))`

Integrate the function:

`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`

Integrate the function:

`1/(x^2(x^4 + 1)^(3/4))`

Integrate the function: 

`1/(x^(1/2) + x^(1/3))  ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))),  "put x" = t^6]`

Integrate the function:

`(5x)/((x+1)(x^2 +9))`

Integrate the function:

`sinx/(sin (x - a))`

Integrate the function:

`(e^(5log x) -  e^(4log x))/(e^(3log x) - e^(2log x))`

Integrate the function:

`cos x/sqrt(4 - sin^2 x)`

Integrate the function:

`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`

Integrate the function:

`1/(cos (x+a) cos(x+b))`

Integrate the function:

`x^3/(sqrt(1-x^8)`

Integrate the function:

`e^x/((1+e^x)(2+e^x))`

Integrate the function:

`1/((x^2 + 1)(x^2 + 4))`

Integrate the function:

`cos^3 xe^(log sinx)`

Integrate the function:

`e^(3log x) (x^4 + 1)^(-1)`

Integrate the function:

f' (ax + b) [f (ax + b)]ⁿ

Integrate the function:

`1/sqrt(sin^3 x sin(x + alpha))`

Integrate the functions `(sin^(-1) sqrtx - cos^(-1) sqrtx)/ (sin^(-1) sqrtx + cos^(-1) sqrtx) , x in [0,1]`

Integrate the function:

`sqrt((1-sqrtx)/(1+sqrtx))`

Integrate the function:

`(2+ sin 2x)/(1+ cos 2x) e^x`

Integrate the function:

`(x^2 + x + 1)/((x + 1)^2 (x + 2))`

Integrate the function:

`tan^(-1) sqrt((1-x)/(1+x))`

Integrate the function:

`(sqrt(x^2 +1) [log(x^2 + 1) - 2log x])/x^4`

Evaluate the definite integral:

`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`

Evaluate the definite integral:

`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx

Evaluate the definite integral:

`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`

Evaluate the definite integral:

`int_(pi/6)^(pi/3)  (sin x + cosx)/sqrt(sin 2x) dx`

Evaluate the definite integral:

`int_0^1 dx/(sqrt(1+x) - sqrtx)`

Evaluate the definite integral:

`int_0^(pi/4) (sin x +  cos x)/(9+16sin 2x) dx`

Evaluate the definite integral:

`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`

Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`

Evaluate the definite integral:

`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`

Prove the following:

`int_1^3 dx/(x^2(x +1)) = 2/3 + log  2/3`

Prove the following:

`int_0^1 xe^x dx = 1`

Prove the following:

`int_(-1)^1 x^17 cos^4 xdx = 0`

Prove the following:

`int_0^(pi/2) sin^3 xdx = 2/3`

Prove the following:

`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`

Prove the following:

`int_0^1sin^(-1) xdx = pi/2 - 1`

Evaluate  `int_0^1 e^(2-3x) dx` as a limit of a sum.

`int dx/(e^x + e^(-x))` is equal to ______.

`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.

If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.

Choose the correct answers The value of `int_0^1 tan^(-1)  (2x -1)/(1+x - x^2)` dx is 

(A) 1

(B) 0

(C) –1

(D) `pi/4`