List of top Mathematics Questions on Some Properties of Definite Integrals

Evaluate : \( \int_{-\pi/2}^{\pi/2} \frac{\cos^2 x}{2^x + 1} \, dx \)

Evaluate the definite integral: \[ \int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{1}{1 + \sqrt{\cot x}} \, dx \]

The value of \[ \int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x^{11}}{1 - \sin\left(|x| + \frac{\pi}{6}\right)} \right) dx \] is equal to

Evaluate \[ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{x\sin x}{1+\cos2x}\,dx \]

Evaluate $\int_{-\pi}^{\pi} \frac{\cos^2 x}{1+a^x} dx$

\[ \int_0^{\frac{\pi}{2}} \frac{\sin^{100}x}{\sin^{100}x+\cos^{100}x}\,dx = \]

The value of the definite integral \( \int_0^2 \frac{1}{3^x + 3} dx \) is

The value of the definite integral $\int_{-\pi/2}^{\pi/2} (x^3 + x\cos x + \tan^5 x + 1) \, dx$ is:

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

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

The value of $\int_{0}^{\frac{\pi}{2}}\frac{\cos^{11}x}{\cos^{11}x+\sin^{11}x}dx$ is equal to} \textit{Note: The original exam paper contained a typographical error ($c\hat{D}$ instead of $dx$). It has been corrected here to permit a valid solution.

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

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

\(6 \int_{0}^{\pi} (\sin 3x + \sin 2x + \sin x)dx\) is equal to:

The number of elements in the set \[ S = \left\{ x : x \in [0,100] \text{ and } \int_{0}^{x} t^2 \sin(x - t)\,dt = x^2 \right\} \] is

Find the area bounded by \( y = \max \{ \sin x, \cos x \} \) when \( x \in \left[ 0, \frac{3\pi}{2} \right] \) with the x-axis:

The value of \[ \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{1}{1 + \sqrt{\tan 2x}} \, dx \] is:

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.

Given that $\int_{0}^{1}\tan^{-1}(t)\,dt = \frac{\pi}{4} - \frac{1}{2}\log 2$, then $\int_{0}^{1}\tan^{-1}(1-t)\,dt =$

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

\(\displaystyle \int_{-a}^{a}|x|\,dx=\)

The value of \(\displaystyle \int_0^1 x(1-x)^9\,dx\) is

\(\displaystyle \int_{-a}^{a}|x|\,dx=\)