Arts (English Medium) Class 12 - CBSE Question Bank Solutions

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`

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, find the angle which `veca + vecb + vecc`make with `veca or vecb or vecc`

Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.

\[\int\frac{1}{x} \left( \log x \right)^2 dx\]