List of top Mathematics Questions on Some Properties of Definite Integrals

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

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

If \(\int _{-π/2}^{π/2}(sin^2x+sin^3x)dx = k\), then the value of \(k\)

If \(\int _0^2x(2-x)^b\,dx = \frac{32}{7}\), where \(b\in N\) then \(b =\)

The value of \(\int _0^π\frac{1}{1+2^{cosx}}\,dx\) is _____

The value of \(\int _{-π}^π\frac{2x(1+sinx)}{1+cos^2x}dx\) is...

\(\int _{π/6}^{π/3}\frac{sinx-cosx}{1+sinxcosx}\,dx =\)

If \(f(x)\) is an even function, then \(\int _{-2}^2(|x|+f(x)sinx)\,\text{d}x\) is

The value of the integral \(\int _{-π/2}^{π/2}(\frac{x^2cosx}{1+e^x})dx\) is equal to \((\frac{π^2}{A})-B\). Then \((\frac{A}{B}) =\)

If \(\int _{π/6}^{π/3}\frac{1}{1+sinx+cosx}dx = log2\), then the value of \(\int _{π/6}^{π/3}\frac{cosx}{1+sinx+cosx}dx =\) ............

The value of the integral \(\int _{-\sqrt{3}}^{\sqrt{3}}\frac{(2x^9+3x^8-5x^7+9x^6+4x^3-x+3)}{x^2+3}\,dx\) is

If \(\int _0^k\frac{1}{1+4^x}dx = log_2(\frac{4}{3})\), then \(k =\) _____

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

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

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

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

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

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: