List of top Mathematics Questions on Integration asked in UP Board XII

Integrate \( \log_e x \) with respect to \( x \).

Prove: \[ \int_0^{\frac{\pi}{4}} \log_e (1 + \tan x) \, dx = \frac{\pi}{8} \log_e 2. \]

Evaluate: \[ \int \frac{x^2 + x + 1}{(x + 2)(x^2 + 1)} \, dx. \]

Evaluate: \[ \int_0^\pi \frac{x \sin x}{1 + \cos^2 x} \, dx. \]

Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x + \cos x}} \, dx. \]

Evaluate: \[ \int \frac{\sec^2(2x)}{(\cot x - \tan x)^2} \, dx. \]

Find the value of \( \int_a^b x^2 \, dx \) with the help of definite integral as the limit of a sum.

The value of \[ \int \cos^2 x \, dx \] will be

Evaluate: \[ \int \sqrt{x^2 - a^2} \, dx. \]

Evaluate: \[ \int \sqrt{x^2 + 2x + 5} \, dx \]

The value of $\int x e^x \, dx$ will be:

Evaluate: \[ \int \left( \sqrt{\cot x} + \sqrt{\tan x} \right) dx \]

Evaluate: \[ \int \tan^4 x \sec^2 x \, dx. \]

The value of \[ \int \frac{dx}{x^2 - a^2} \] will be:

Prove that \[ \int \sin^{-1}\!\left(\frac{2x}{1+x^2}\right) dx = 2x \tan^{-1}x - \log(1+x^2) + C. \]

Evaluate: \[ \int \sqrt{x^2 - 8x + 7}\, dx \]

Evaluate: \[ \int \sec x \, (\sec x + \tan x) \, dx \]

Evaluate: \[ \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \]

Evaluate: $\int x^{2}\sin(x^{3})\,dx$