List of top Mathematics Questions on Definite Integral asked in MHT CET

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:

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

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

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

The value of \( \displaystyle \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \) is:

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

Evaluate the definite integral: \[ \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx \]

Evaluate the integral: \[ \int_{1}^{4}\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{e^x}}\right)dx \]

Evaluate the integral: \[ \int_{3}^{5} |x-4|\,dx \]

Evaluate the integral \[ \int_0^{\pi/2} \frac{\cos^2 x \sin^2 x}{\cos^2 x + \sin^2 x} \, dx. \]

Evaluate the integral
\[ \int_{-1}^1 \left( \tan^{-1} \left( \frac{x}{x+1} \right) + \tan^{-1} \left( \frac{x+1}{x} \right) \right) \, dx \]

$\int_{\log(1/2)}^{\log 2} \sin \left( \frac{e^x - 1}{e^x + 1} \right) dx = \_\_\_\_\_\_.$

$\int_{\pi/4}^{\pi/2} 2\sin^{-4} x dx = \_\_\_\_\_\_.$
Note: The initial OCR showed "$23.4 \frac{/2}{/4}$". The "4" was a misread coefficient. The mathematical evaluation of the options indicates a coefficient of 2 is present in the intended question.

$\int_0^1 \tan^{-1} x \, dx = $ ______.

$\int_{1/2}^{2} \frac{1}{x} \csc^{101} \left( x - \frac{1}{x} \right) dx = $ ______.

$\int_{0}^{\frac{\pi}{2}}\frac{300 \sin x+100 \cos x}{\sin x+\cos x}dx=...$

$\int_{0}^{1}x|x-\frac{1}{2}|dx=$

The value of \[ \int_{0}^{\pi} \left|\sin^{3}x\right| \, dx \] is:

Evaluate the definite integral. $$ \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \frac{x}{1+\sin x} \, dx = $$

( \int_0^1 \log(x + 1) , dx = )

The value of ( \int_0^\pi |\sin^3 x| , dx ) is