NCERT solutions for Class 12 Maths chapter 7 - Integrals [Latest edition]

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

sin 2x

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

Cos 3x

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

`e^(2x)`

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

(ax + b)²

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

sin 2x – 4 e^3x

Find the following integrals ∫(4 e^3x + 1) dx

Find the following integrals `intx^2 (1 - 1/x^2)dx`

Find the following integrals ∫(ax + 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 ∫(1 – x) `sqrtx` dx

Find the following integrals ∫ x( 3x² + 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 ∫sec 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

 (A) `1/3 x^(1/3) + 2x^(1/2) + C`

(B) `2/3x^(2/3) + 1/2 x^2 + C`

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

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

if `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0 Then f(x) is

(A) `x^4 + 1/x^3 - 129`

(B)`x^3 + 1/x^4 + 129/8`

(C) `x^4 + 1/x^3 + 129/8`

(D) `x^3 + 1/x^4 - 129/8`

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`

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^23 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 in `(sin x)/(1+ cos x)^2`

Integrate the functions in `1/(1 + cot x)`

Integrate the functions in `1/(1 - tan x)`

Integrate the functions in `sqrt(tanx)/(sinxcos x)`

Integrate the functions in `(1+ log x)^2/x`

Integrate the functions in `((x+1)(x + logx)^2)/x`

Integrate the functions in `(x^3 sin(tan^(-1) x^4))/(1 + x^8)`

Choose the correct answer int `(10x^9 + 10^x log_e 10)/(x^10 + 10^x)` dx equals

(A) 10^x – x¹⁰ + C

(B) 10x + x¹⁰ + C

(C) (10^x – x¹⁰)–1 + C

(D) log (10^x + x¹⁰) + C

Choose the correct answer `int (dx)/(sin^2 x cos^2 x)` equals

(A) tan x + cot x + C

(B) tan x – cot x + C

(C) tan x cot x + C

(D) tan x – cot 2x + C

Find the integrals of the functions sin² (2x + 5)

Find the integrals of the functions sin 3x cos 4x

Find the integrals of the functions cos 2x cos 4x cos 6x

Find the integrals of the functions sin³ (2x + 1)

Find the integrals of the functions sin³ x cos³ x

Find the integrals of the functions sin x sin 2x sin 3x

Find the integrals of the functions sin 4x sin 8x

Find the integrals of the functions `(1- cosx)/(1 +  cos x)`

Find the integrals of the functions  `cos x/(1 + cos x)`

Find the integrals of the functions `sin^4 x`

Find the integrals of the functions cos⁴ 2x

Find the integrals of the functions `(sin^2 x)/(1 + cos x)`

Find the integrals of the functions  `(cos 2x - cos 2 alpha)/(cos x - cos alpha)`

Find the integrals of the functions  `(cos x -  sinx)/(1+sin 2x)`

Find the integrals of the functions tan³ 2x sec 2x

Find the integrals of the functions tan⁴x

Find the integrals of the functions `(sin^3 x + cos^3 x)/(sin^2x cos^2 x)`

Find the integrals of the functions `(cos 2x+ 2sin^2x)/(cos^2 x)`

Find the integrals of the functions `1/(sin xcos^3 x)`

Find the integrals of the functions `(cos 2x)/(cos x + sin x)^2`

Find the integrals of the functions sin⁻¹ (cos x)

Find the integrals of the functions  `1/(cos(x - a) cos(x - b))`

Choose the correct answer `int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx`

A. tan x + cot x + C

B. tan x + cosec x + C

C. − tan x + cot x + C

D. tan x + sec x + C

Choose the correct answer `int (e^x(1 +x))/cos^2(e^x x) dx`

A. − cot (e^xx) + C

B. tan (xe^x) + C

C. tan (e^x) + C

D. cot (e^x) + C

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Choose the correct answer int `dx/(x^2 + 2x + 2)` equals

A. x tan⁻¹ (x + 1) + C

B. tan^{− 1} (x + 1) + C

C. (x + 1) tan⁻¹ x + C

D. tan⁻¹ x + C

Choose the correct answer `int dx/sqrt(9x - 4x^2)` equals

(A) `1/9 sin^(-1) ((9x -8)/8) + C`

(B) `1/2 sin^(-1) ((8x -9)/9) + C`

(C) `1/3 sin^(-1) ((9x - 8)/8) + C`

(D) `1/2 sin^(-1) ((9x - 8)/9) + C`

Integrate the rational functions `x/((x +  1)(x+ 2))`

Integrate the rational functions `1/(x^2 - 9)`

Integrate the rational functions `(3x - 1)/((x - 1)(x - 2)(x - 3))`

Integrate the rational functions `x/((x-1)(x- 2)(x - 3))`

Integrate the rational functions `(2x)/(x^2 + 3x + 2)`

Integrate the rational functions `(1 - x^2)/(x(1-2x))`

Integrate the rational functions `x/((x^2+1)(x - 1))`

Integrate the rational functions `x/((x -1)^2 (x+ 2))`

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

Integrate the rational functions `(2x - 3)/((x^2 -1)(2x + 3))`

Integrate the rational functions `(5x)/((x + 1)(x^2 - 4))`

Integrate the rational functions `(x^2 + x + 1)/(x^2 -1)`

Integrate the rational functions `2/((1-x)(1+x^2))`

Integrate the rational functions `(3x -1)/(x + 2)^2`

Integrate the rational functions `1/(x^4 - 1)`

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

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

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

Integrate the rational functions  `(2x)/((x^2 + 1)(x^2 + 3))`

Integrate the rational functions `1/(x(x^4 - 1))`

Integrate the rational functions `1/(e^x -1)`[Hint: Put e^x = t]

Choose the correct answer `int (xdx)/((x - 1)(x - 2))` equals

A. `log |(x - 1)^2/(x -2)| + C`

B. `log |(x-2)^2/(x -1)| +C`

C. `log|((x- 1)/(x- 2))^2| + C`

D. log|(x - 1)(x - 2)| + C

Choose the correct answer `int (dx)/(x(x^2 + 1))` equal

A. `log |x| - 1/2 log (x^2 + 1) + C`

B. `log |x| + 1/2 log(x^2 + 1) + C`

C. `- log|x| + 1/2 log (x^2 + 1) + C`

D. `1/2 log|x| + log(x^2 + 1)+ C`

Integrate the functions x sin x

Integrate the functions x sin 3x

Integrate the functions `x^2e^x`

Integrate the functions x logx

Integrate the functions x log 2x

Integrate the functions x² log x

Integrate the functions x sin^{– 1}x

Integrate the functions x tan^–1 x

Integrate the functions x cos^–1 x

Integrate the functions (sin^–1x)²

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

Integrate the functions x sec² x

Integrate the functions tan^–1x

Integrate the functions x (log x)²

Integrate the functions (x² + 1) log x

Integrate the functions e^x (sinx + cosx)

Integrate the functions `(xe^x)/(1+x)^2`

Integrate the functions `e^x (1 + sin x)/(1+cos x)`

Integrate the functions `e^x (1/x - 1/x^2)`

Integrate the functions `((x- 3)e^x)/(x - 1)^3`

Integrate the functions e^2x sin x

Integrate the functions `sin^(-1) ((2x)/(1+x^2))`

Choose the correct answer `intx^2 e^(x^3) dx` equals

(A) `1/3 e^(x^3) + C`

(B) `1/3 e^(x^2) + C`

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

(D) `1/3 e^(x^2) + C`

Choose the correct answer `int e^x sec x (1 +   tan x) dx` equals

(A) e^x cos x + C

(B) e^x sec x + C

(C) e^x sin x + C

(D) e^x tan x + C

Integrate the functions `sqrt(4 - x^2)`

Integrate the functions `sqrt(1- 4x^2)`

Integrate the functions `sqrt(x^2 + 4x + 6)`

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

Integrate the functions `sqrt(1-4x - x^2)`

Integrate the functions `sqrt(x^2 + 4x - 5)`

Integrate the functions `sqrt(1+ 3x - x^2)`

Integrate the functions `sqrt(x^2 + 3x)`

Integrate the functions `sqrt(1+ x^2/9)`

Choose the correct answer `int sqrt(1+ x^2) dx` is equal

(A) `x/2 sqrt(1+ x^2) +  1/2log |(x + sqrt(1+ x^2)) + C|`

(B) `2/3 (1+ x^2)^(3/2) + C` 

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

(D) `x^2/2 sqrt(1+x^2) + 1/2 x^2 log|x + sqrt(1+ x^2)|+ C`

Choose the correct answer `int sqrt(1+ x^2) dx` is equal

(A) `x/2 sqrt(1+ x^2) +  1/2log |(x + sqrt(1+ x^2)) + C|`

(B) `2/3 (1+ x^2)^(3/2) + C` 

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

(D) `x^2/2 sqrt(1+x^2) + 1/2 x^2 log|x + sqrt(1+ x^2)|+ C`

Choose the correct answer `int sqrt(x^2 - 8x + 7) dx` is equal to

(A) `1/2 (x - 4) sqrt(x^2 - 8x + 7) + 9log |x - 4 +  sqrt(x^2 - 8x + 7)| + C`

(B) `1/2( x- 4) sqrt(x^2-8x + 7) + 9log|x + 4 + sqrt(x^2 - 8x + 7)| + C`

(C) `1/2 (x - 4) sqrt(x^2 - 8x + 7) - 3sqrt2 log |x - 4 + sqrt(x^2 - 8x + 7)| + C`

(D) `1/2 (x - 4) sqrt(x^2 - 8x + 7) - 9/2 log |x - 4 +  sqrt(x^2 - 8x + 7)|  + C`

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

Evaluate the definite integrals `int_2^3 1/x dx`

Evaluate the integrals in using substitution

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

Evaluate the definite integrals  `int_1^2 (4x^3 - 5x^2 + 6x + 9)`                     

Evaluate the definite integrals `int_0^(x/4) sin2xdx`

Evaluate the definite integrals `int_0^(x/2) xos 2x dx`

Evaluate the definite integrals `int_4^5 e^x dx`

Evaluate the definite integrals `int_0^(x/4) tan x dx`

Evaluate the definite integrals `int_(pi/6)^(pi/4) cosec x dx`

Evaluate the definite integrals `int_0^4 dx/sqrt(1-x^2)`

Evaluate the definite integrals `int_0^4 dx/(1+x^2)`

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

A. cos x + x sin x

B. x sin x

C. x cos x

D. sin x + x cos x

Evaluate the definite integrals `int_2^3 dx/(x^2 - 1)`

Evaluate the definite integrals `int_0^(pi/2) cos^2 xdx`

Evaluate the definite integrals `int_2^3 (xdx)/(x^2 + 1)`

Evaluate the definite integrals `int_0^1 (2x + 3)/(5x^2 + 1) dx`

Evaluate the definite integrals `int_0^1 x e^(x^2) dx`

Evaluate the definite integrals `int_1^2 (5x^2)/(x^2 + 4x + 3)`

Evaluate the definite integrals `int_0^(pi/4) (2 sec^2 x + x^3  + 2) dx`

Evaluate the definite integrals `int_0^Pi (sin^2 x/2 - cos^2 x/2) dx`

Evaluate the definite integrals `int_0^2 (6x +3)/(x^2 + 4)` dx

Evaluate the definite integrals `int_0^1 (xe^x + sin  (pix)/4)`

Choose the correct answer `int_1^(sqrt3)dx/(1+x^2) ` equals

A. `pi/3`

B. `(2pi)/3`

C. `pi/6`

D. `pi/12`

Choose the correct answer `int_0^(2/3) dx/(4+9x^2)` equals is 

Evaluate the integrals in using substitution

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

Evaluate the integrals in using substitution 

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

Evaluate the integrals in using substitution

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

Evaluate the integrals in using substitution

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

Evaluate the integrals in using substitution

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

Evaluate the integrals in using substitution

 `int_(-1)^1 dx/(x^2 + 2x  + 5)`

Evaluate the integrals in 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

A. 6

B. 0

C. 3

D. 4

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

`int_2^8 |x - 5| dx`

By using the properties of definite integrals, evaluate the integrals

`int_0^4 x(1-x)^n dx`

By using the properties of definite integrals, evaluate the integrals 

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals 

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

`int_0^pi (xdx)/(1+ sin x)`

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

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

By using the properties of definite integrals, evaluate the integrals

`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`

Choose the correct The value of `int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is

A. 0

B. 2

C. π

D. 1

Choose the correct The value of `int_0^(pi/2) log  ((4+ 3sinx)/(4+3cos x))` dx is

A. 2

B. 3/4

C. 0

D. -2

Integrate the functions `1/(x - x^3)`

Integrate the functions `1/(sqrt(x+a) + sqrt(x+b))`

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

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

Integrate the functions 

`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 functions  `(5x)/((x+1)(x^2 +9))`

Integrate the functions `sinx/(sin (x - a))`

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

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

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

Integrate the functions `1/(cos (x+a) cos(x+b))`

Integrate the functions `x^3/(sqrt(1-x^8)`

Integrate the functions  `e^x/((1+e^x)(2+e^x))`

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

Integrate the functions `cos^3 xe^(log sinx)`

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

Integrate the functions `f'(ax +b)[f(ax +b)]^n`

Integrate the functions `1/sqrt(sin^3 x sin(x + a))`

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

Integrate the functions `sqrt((1-sqrtx)/(1+sqrtx))`

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

Integrate the functions `(x^2 + x + 1)/((x + 1)^2 (x + 2))`

Integrate the functions `tan^(-1) sqrt((1-x)/(1+x))`

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

Evaluate the definite integrals `int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`

Evaluate the definite integrals `int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx

Evaluate the definite integrals  `int_0^(pi/2) (cos^2 x dx)/(xos^2 x + 4 sin^2 x)`

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

Evaluate the definite integrals `int_0^1 dx/(sqrt(1+x) - sqrtx)`

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

Evaluate the definite integrals `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 integrals `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_0^4 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.

Choose the correct `int dx/(e^x + e^(-x))` is equal to

(A) tan^–1 (e^x) + C

(B) tan^–1 (e^–x) + C

(C) log (e^x – e^–x) + C

(D) log (e^x + e^–x) + C

Choose the correct answers `int (cos 2x)/(sin x + cos x)^2dx` is equal to

A `(-1)/(sin x + cos x) + C`

(B) log |sin x + cos x | + C

(B) log |sin x + cos x | + C

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

Choose the correct answers If f (a + b – x) = f (x), then `int_a^b x f(x )dx` is equal

A `(a+b)/2 int_a^b f(b-x) dx`

B  `(a+b)/2 int_a^b f(b+x) dx`

C `(b-a)/2 int_a^b f(x) dx`

D `(a+b)/2 int_a^b f(x) dx`

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`