Integration

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 area of the plane bounded by $y=\sqrt{x},\ x\in[0,1]$, $y=x^2,\ x\in[1,2]$, $y=-x^2+2x+4,\ x\in[0,2]$

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

The smaller area bounded by $y = 2 - x$ and $x^2 + y^2 = 4$ is:

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

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

If $ f(x)=\displaystyle \int_{0}^{x} t\sin t, dt $, then $f'(x)$ 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

The value of $\int \frac{(x+1)}{x(xe^{x}+1)} dx$ is equal to

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

If $\int e^{x}(f(x)-f'(x))dx=\phi(x)$ , then the value of $\int e^x f(x) dx$ is

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