Integrals Test ...

The value of the integral $$\displaystyle \int_{0}^{\pi /4}\displaystyle \frac{\sin x+\cos x}{3+\sin 2x}dx$$ is

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

If $$I= \displaystyle \int_{1}^{2}\displaystyle \frac{dx}{\left ( x+1 \right )\sqrt{x^{2}-1}}$$ then $$I$$ is equal to

The value of $$\displaystyle \int_{0}^{1}\displaystyle \frac{dx}{e^{x}+e^{-x}}$$ is

The value of $$\displaystyle \int_{8}^{15}\displaystyle \frac{dx}{\left ( x-3 \right )\sqrt{x+1}}$$ is

If $$b > a,$$ and $$\displaystyle  I = \int _{a}^{b} \sqrt{\frac{ x-a}{b-x} }dx,$$ then $$I$$ equals

f $$ k = e^{2007} $$ then value of $$ \displaystyle I =\int_{1}^{k}\frac{ \pi \cos (\pi \log x )} {x} dx $$ is

Value of $$\displaystyle \int_{0}^{25}\displaystyle \frac{1}{\sqrt{4+\sqrt{x}}}\: dx$$ is

$$\displaystyle \int_{0}^{\infty }f\left ( x+\frac{1}{x} \right )\frac{\ln x}{x}dx$$

If$$\displaystyle \int_{0}^{\pi /3}\frac{\cos }{3+4\sin x}dx=K\log \frac{\left ( 3+2\sqrt{3} \right )}{3}$$ then K is