Definite Integrals

If $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ are continuous functions, then evaluate $\displaystyle \int_{-\pi/2}^{\pi/2}[f(x)+f(-x)][g(x)-g(-x)],dx$

The value of $\displaystyle \int_0^{\pi/2} \frac{dx}{1+\tan^3 x}$ is

The value of $\displaystyle \int_{0}^{\pi} \frac{x \sin x}{1+\cos^2 x},dx$ is:

If area between $ y=x^{2} $ and $ y=x $ is $ A $, then area between $ y=x^{2} $ and $ y=1 $ is:

$ \displaystyle \int_{0}^{1/2} \frac{dx}{\sqrt{x - x^{2}}} $

If $a$ is a positive integer, then the number of values satisfying $ \displaystyle \int_{0}^{\pi/2} \left[ a^{2}\left(\frac{\cos 3x}{4}+\frac{3}{4}\cos x\right)+a\sin x - 20\cos x \right] dx \le -\frac{a^{2}}{3} $ is

The value of the integral $\int _0^{\pi/2} \frac{\sqrt{sinx}}{\sqrt{sinx}+\sqrt{cosx}} dx$ is

If $I_n = \int_0^{\pi/4} tan^{n} \theta d\theta$ , then $I_8 + I_6$ equals

If [x] represents the greatest integer not exceeding x, then $\int_{0}^{9} [x] dx $ is

The value of $\int_{0}^{\pi/4} log(1+tanx)dx$ is equal to:

$\int_0^\pi [cotx]dx$ where [.] denotes the greatest integer function, is equal to

The value of $\int_{-\pi/3}^{\pi/3} \frac{x sinx}{cos^{2}x}dx$

Evaluate $\displaystyle \int_{0}^{1}x(1-x)^ndx $

The value of $\int_{0}^{\pi}x^3 \sin x dx$

Find the area bounded by the line y = 3 - x, the parabola y = x² - 9 and

The value of  depends on the

If  where n is a positive integer, then the relation between I_n and I_n-1 is

Evaluate

The area of the region bounded by x-axis and the curves defined by $y=tanx$, $-\frac{\pi}{3}\leq x\leq \frac{\pi}{3}$ and $y=cotx$, $-\frac{\pi}{6}\leq x\leq \frac{3\pi}{2}$ is

The value of $\displaystyle \int_{0}^{\sin^2 x} \sin^{-1}\sqrt{t} dt + \int_{0}^{\cos^2 x} \cos^{-1}\sqrt{t} dt$ is:

The value of integral $\displaystyle \int_{0}^{\pi/2} \log \tan x dx$ is

If $ I_1 = \displaystyle \int_{0}^{1} 2^{x^2},dx,\quad I_2 = \displaystyle \int_{0}^{1} 2^{x^3},dx,\quad I_3 = \displaystyle \int_{1}^{2} 2^{x^2},dx,\quad I_4 = \displaystyle \int_{1}^{2} 2^{x^3},dx,$ then

The value of $\displaystyle \lim_{n\to\infty} \frac{\pi}{n}\left[\sin\frac{\pi}{n}+\sin\frac{2\pi}{n}+\cdots+\sin\frac{(n-1)\pi}{n}\right]$ is:

The value of $\int ^{\pi/3}_{-\pi/3}\frac{x\sin x}{{\cos }^2x}dx$ is

Which of the following is TRUE?

The value of $\int ^{\frac{\pi}{2}}_0\frac{(1+2\cos x)}{({2+\cos x)}^2}dx$ lies in the interval

The area enclosed between the curve y = sin x, y = cosx, $0\leq x\leq\frac{\pi}{2}$ is

The value of $\frac{d}{dx}\int ^{2\sin x}_{\sin {x}^2}{e}^{{t}^2}dt$ at $x=\pi$