List of top Mathematics Questions on Some Properties of Definite Integrals asked in MHT CET

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

Evaluate the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^{101}} \]

If \( 0 \le \cos^{-1} x \le \pi \) and \( -\frac{\pi}{2} \le \sin^{-1} x \le \frac{\pi}{2} \), then at \( x = \frac{1}{5} \) the value of \( \cos(2 \cos^{-1} x + \sin^{-1} x) \) is

The volume of the tetrahedron whose co-terminus edges are $\bar{a}, \bar{b}, \bar{c}$ is 12 cubic units. If the scalar projection of $\bar{a}$ on $\bar{b} \times \bar{c}$ is 4 , then $|\bar{b} \times \bar{c}| =$

The distance of the point $(-3, 2, 3)$ from the line passing through $(4, 6, -2)$ and having direction ratios $-1, 2, 3$ is ________ units.

The normal to the curve $x = 9(1 + \cos \theta), y = 9 \sin \theta$ at $\theta$ always passes through the fixed point}

Evaluate the integral: $\int_0^{\pi / 2} \frac{\cos x}{3 \cos x + \sin x} dx$

Evaluate the integral: $\int_{0}^{1} |5x - 3| dx$

Evaluate the definite integral $$\int_{1}^{3} \left[ \tan^{-1}\left(\frac{x}{x^2-1}\right) + \tan^{-1}\left(\frac{x^2-1}{x}\right) \right] dx$$

$\int_{0}^{\pi / 2} \log\left(\frac{4+3\sin x}{4+3\cos x}\right) dx = $

$\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1-\sin x \cos x} d x=$

Evaluate the definite integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{1 + e^x}\ dx$$

If $\int_2^e \left[ \frac{1}{\log x} - \frac{1}{(\log x)^2} \right] dx = a + \frac{b}{\log 2}$, then

If $2 f(x) - 3 f\left(\frac{1}{x}\right) = x$, then $\int_1^e f(x) dx =$

$\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1-\sin x \cos x} d x=$