List of top Mathematics Questions on Definite Integral

The value of the definite integral $\int_0^{\pi/2} \ln(\tan x) \, dx$ is:

The value of the definite integral $\int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx$ is:

Let \[ f(x)=\int_{1}^{4}\log[x]\ dx, \] where \([x]\) denotes the greatest integer function. Then the value of \(f(x)\) is:

If \(n\) is an odd natural number and \[ I_n=\int_0^1 e^x(x-1)^n\,dx \] then \( I_n+nI_{n-1} \) is equal to:

Evaluate: \[ \int_0^1 x^3\ln(1+x)\,dx \]

Evaluate the definite integral: \( \int_{0}^{\pi/2} \sin^2(x) \, dx \).

Evaluate the integral \[ \int_{0}^{\frac{\pi}{2}} \frac{3\sin x+4\cos x}{\sin x+\cos x}\,dx \]

If \( f(x) = \int_0^x t(\sin x - \sin t)dt \) then :

The value of $\int_{0}^{4} \sqrt{\frac{4-x}{4+x}} dx$ is:

Evaluate the definite integral: \( \displaystyle \int_{3}^{5} |x-4|\,dx \).

Evaluate the integral \( \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \).

Find the value of the integral \( \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \).

Evaluate \( \displaystyle \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \).

Find the value of \( \displaystyle \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx \).

$$ \int_{0}^{\pi/4} (\tan^8 x + \tan^6 x) \, dx = ? $$

$$ \int_{-\pi/4}^{\pi/4} \sin^{103} x \cos^{101} x \, dx = ? $$

One of the possible functions $f(x)$ which satisfies $\int_{-2}^{2} f(x) dx = 0$ is

$\int_{a-6}^{b-6} f(x + 6) dx$ is equal to

The value of $\int_{0}^{3}x^{2}[x]dx$ is equal to

The value of $\int_{-2}^{1}\frac{|x|}{x}dx$ is equal to

The value of $\int_{0}^{\pi}\frac{\sin x}{1+\sin x}dx$ is equal to} \textit{Note: The upper limit in the integral from the original paper contains a typo ($x$ instead of $\pi$). It has been corrected here to yield the valid options provided.

The value of $\displaystyle \int_{0}^{1} x(1-x)^4 \, dx$ is equal to:

$\displaystyle \int_{-6}^{0} \left[t^3 + 9t^2 + 27t + 29 + (t+3)\cos(t+3)\right] dt$ is equal to:

If $I = \displaystyle \int_{-1}^{1} \frac{x^4}{1 - x^4} \cos^{-1}\left(\frac{2x}{1+x^2}\right) dx$, then $2I$ is equal to: