List of top Mathematics Questions on Integration

(b) The value of \( \int x \sin x \, dx \) will be:

(b) Find the value of: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]

(a) Prove that: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \sin x}{1 + \cos^2 x} \, dx = \frac{\pi^2}{4}. \]

(c) Solve: \[ \int \frac{3x + 5}{x^3 - x^2 - x + 1} \, dx. \]

(c) The value of the integral: \[ \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{dx}{1+x^2} \]

The value of \( \int \sin^2 x \, dx \) is:

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: \[ \int \frac{2x^2 - 3}{(x^2 - 5)(x^2 + 4)} \, dx \]

Prove that: \[ \int_0^a f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx \]

Evaluate: \[ \int x \tan^{-1}{x} \, dx \]

Evaluate: \[ \int_0^{\frac{\pi}{2}} \sqrt{1 - \cos{4x}} \, dx \]

Write \( \int \cot{x} \, dx \).

The integral of \( \cos^2x \) is

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