List of top Mathematics Questions on Definite Integrals

Evaluate the integral \[ \int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx . \]

Evaluate: \(\displaystyle\int_0^2 \left([x^2]-[x]^2\right)dx\), where \([\cdot]\) denotes the greatest integer (floor) function.

Consider the improper integrals \(I_\lambda = \displaystyle\int_0^1 \dfrac{dx}{(1-x)^\lambda}\) and \(J_\lambda = \displaystyle\int_1^\infty \dfrac{dx}{x^\lambda}\), \(\lambda \in \mathbb{R}\), and the statements:
(I) For \(\lambda = 1\), \(I_\lambda\) converges, but \(J_\lambda\) diverges.
(II) For \(\lambda = 2\), \(I_\lambda\) diverges, but \(J_\lambda\) converges.
Choose the correct option.

The value of \(\displaystyle\int_0^{\pi/2} \sin^2(x)\cos^4(x)\,dx\) is ____.

Find the value of \[ \int_0^{\pi/2} x \cos x \, dx. \]

Evaluate: \[ \int_0^{\pi/2} \log (\cos x) \, dx. \]

Find the value of \[ \int_0^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}}. \]

Find the value of \[ \int_0^a \sqrt{a^2 - x^2} \, dx. \]

Evaluate: \(\displaystyle \int_0^{\frac{\pi}{6}} \cos x \cdot \cos 2x \, dx = \)

Evaluate: \(\displaystyle \int_{-\pi}^{\pi} \tan x \, dx = \)

Evaluate: \(\displaystyle \int_0^4 \sqrt{x} \, dx = \)

Evaluate: \(\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x \, dx = \)

Evaluate: \(\displaystyle \int_0^a \left( \sqrt{x} + \sqrt{a - x} \right) \sqrt{x} \, dx = \)

Evaluate: \(\displaystyle \int_0^{\frac{\pi}{2}} \cos 2x \, dx = \)

Evaluate: \[ \int_{-1}^{3/2} |x \sin (\pi x)| \, dx. \]