List of top Mathematics Questions on Integration

The value of \( \displaystyle \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be
What is the value of \(\int \frac{1}{x^2 + 1} \, dx\)?
The value of \(\int e^x(\sin x+\cos x)\,dx\) is:
The area bounded by the curve \(y = x^2\), the x-axis and the lines \(x = 1\) and \(x = 2\) is:
The value of \(\displaystyle \int_{-\infty}^{\infty} e^{-x^2}\,dx\) is:
The value of \( \int_0^{\pi/2} \sin^2 x \, dx \) is:

Find the value of \[ \int \frac{\sec^2 2x}{(\cot x - \tan x)^2} \, dx. \] 

(c) If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A \mid B) = \frac{2}{5} \), then find \( P(A \cap B) \).
If \[ \int \frac{3e^x + 5e^{-x}}{4e^x - 5e^{-x}} \, dx = Ax + B \ln |4e^{2x} - 5| + C \] then, the values of $ A $, $ B $, and $ C $ are:
Evaluate the integral: \[ \int_0^2 |x^2 + x - 2| \, dx \]
(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:
Prove that a relation \( R = \{(a, b) : (a-b) \) is a multiple of \( 5 \} \) is an equivalence relation in the set of integers \( \mathbb{Z} \).

Solve:

\[ \int \frac{\sin x}{\sin (x+a)} \, dx. \]

If

\[ A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & -2 \\ -2 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -15 & 5 \\ -1 & 6 & -2 \\ 1 & -5 & 2 \end{bmatrix}, \]

then find the value of \( (AB)^{-1} \).

(d) Find the value of \( \int \log x \, dx \):

Prove that \[ \int_0^{\pi} \frac{x \tan x}{\sec x + \tan x} \, dx = \frac{\pi}{2} (\pi - 2). \] 

Find the value of \[ \int e^x \left( \tan^{-1} x + \frac{1}{1 + x^2} \right) dx. \] 

Integrate \[ \int \frac{\sin(\tan^{-1} x)}{1 + x^2} \, dx. \] 

Find the value of the integral: \[ I = \int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx. \]
Find the value of the integral: \[ I = \int \frac{x^2 + x + 1}{(x+2)(x^2 + 1)} dx. \]
Find the value of \( \int \frac{x+2}{2x^2+6x+5} \,dx \).
Find the value of \( \int x^2 e^{x^3} dx \).
There are 4 red and 4 black balls in a bag and another bag contains 2 red and 6 black balls. One bag is randomly selected and a red ball is drawn. Find the probability that the red ball is drawn from the first bag.
Find the value of \[ \int_0^{\pi/2} \frac{x}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx. \]
Minimize \( Z = 3x + 9y \) under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]