Definite Integration - Calculus - Mathematics Previous Year Questions

If \(k\int\limits_0^1 {x\,.\,f(3x)dx = \int\limits_0^3 {t\,.\,f(t)dt} }\), then the value of \(k\) is

The value of the integral \(\int\limits_0^{\pi /2} {({{\sin }^{100}}x - {{\cos }^{100}}x)dx}\) is

$$\int_{ - \pi /2}^{\pi /2} {\sin xdx}$$

$$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\cos x\,dx}$$

$$\int_0^2\left|x^2+2 x-3\right| d x \text { is equal to }$$

$$\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\cos x-\cos ^3 x} d x \text { is equal to }$$

$$\int_\limits0^{\frac{\pi}{2}} \frac{\cos x}{1+\cos x+\sin x} d x=$$

If \(a\) is a real number such that \(\int_\limits0^a x d x \leq a+4\) then

$$\text { The value of the integral } \int_\limits{\frac{1}{3}}^1 \frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4} d x \text { is }$$

$$\text { If } I_n=\int_\limits0^{\frac{\pi}{4}} \tan ^n x d x \text {, for } n \geq 2 \text {, then } I_n+I_{n-2}=$$

$$\int_\limits{-1}^1 \frac{d}{d x}\left(\tan ^{-1} \frac{1}{x}\right) d x \text { is }$$

\(\int\limits_{-\pi / 2}^{\pi / 2} \sin ^2 x d x\) is equal to