Integrals Test ...

If $$\displaystyle I_{n}= \int_{0}^{\pi /4}\tan ^{n}x\times \sec ^{2}x dx,$$ then $$\displaystyle I_{1}, I_{2}, I_{3},$$ .......are in

The value of $$\displaystyle \int ^{\tan x}_{1/e}\displaystyle \frac{t\, dt}{1+t^{2}}+\displaystyle \int ^{\cot x}_{1/e}\displaystyle \frac{dt}{t\left ( 1+t^{2} \right )}$$ is

$$\displaystyle \int_{\pi /6}^{\pi /4}\frac{dx}{\sin 2x}$$is equal to

The value of $$\displaystyle \int_{1/e}^{\tan x}\displaystyle \frac{t}{1+t^{2}}\, dt\, +\, \displaystyle \int_{1/e}^{\cot x}\displaystyle \frac{dt}{t\left ( 1+t^{2} \right )}$$ is

If $$0< \alpha < \pi /2$$ then the value of $$\displaystyle \int_{0}^{\alpha }\displaystyle \frac{dx}{1-\cos x\cos \alpha }$$ is

If $$\displaystyle \int_{\log 2}^{x}\displaystyle \frac{dx}{\sqrt{e^{x}-1}}= \displaystyle \frac{\pi }{6}$$, the value of $$x$$ is

If $$I= \displaystyle \int_{0}^{1/\sqrt{3}}\displaystyle \frac{dx}{\left ( 1+x^{2} \right )\sqrt{1-x^{2}}}$$ then $$I$$ is equal to

The value of the integral $$\displaystyle \int_{\alpha }^{\beta }\displaystyle \dfrac{dx}{\sqrt{\left ( x-\alpha  \right )\left ( \beta -x \right )}}$$ for $$\beta > \alpha $$, is

The value of $$\displaystyle \int_{-4}^{-5}e^{\left ( x+5 \right )^{2}}dx+3\displaystyle \int_{1/3}^{2/3}e^{9\left ( x-2/3 \right )^{2}}dx$$ is

If $$\displaystyle \int _{ 0 }^{ 1 }{ \frac { \sin { t }  }{ 1+t } dt } =\alpha $$, then the value of the integral $$\displaystyle \int _{ 4\pi -2 }^{ 4\pi  }{ \frac { \sin { t/2 }  }{ 4\pi +2-t } dt } $$ in terms of $$\alpha$$ is given by