Integrals Test ...

If $$\displaystyle I_{t}=\int_{0}^{\dfrac{\pi }{2}}\frac{\sin^{2}tx}{\sin^{2}x}dx$$ then ,$$I_{1},I_{2},I_{3}$$ are in

Let $$\displaystyle F\left ( x \right )=f\left ( x \right )+f\left ( \frac{1}{x} \right )$$ where $$\displaystyle f\left ( x \right )=\int_{1}^{x}\frac{\log t}{1+t}dt$$ 

The value of  $$\displaystyle \int_{\frac{\pi }{5}}^{\frac{3\pi }{10}}\dfrac{\cos x}{\sin x+\cos x} dx$$ equals

The value of $$\displaystyle \int _{ 0 }^{ \pi /2 }{ \frac { d\theta  }{ 5+3\cos { \theta  }  }  }$$ is?

$$\displaystyle \int_{0}^{\pi /2}\sin x\log \left ( \sin x \right )dx= $$

If $$\displaystyle I_1=\int_{0}^{\pi /2}\frac{x}{\sin x}dx $$ and $$\displaystyle I_2=\int_{0}^{\pi /2}\frac{\tan ^{-1}x}{x}dx, $$ then $$\displaystyle \frac{I_{1}}{I_{2}}= $$ 

$$\displaystyle If \int_{-1}^{1}\frac{g\left ( x \right )}{1+t^{2}}dt= f\left ( x \right ) , where,  g\left ( x \right )= \sin x$$ , then $$ {f}'\left ( \frac{\pi }{3} \right )$$ equals

$$\displaystyle\int_{0}^{a}x^{4}\left ( a^{2}-x^{2} \right )^{1/2} dx$$ equals

Evaluate : $$\displaystyle \int_{-\frac{1}{\sqrt2}}^{\frac{1}{\sqrt2}}\frac{x^{8}}{1-x^{4}}\times \left [ \sin ^{-1}\left ( 1-2x^{2} \right ) +\cos ^{-1}\left ( 2x\sqrt{1-x^{2}} \right )\right ]dx$$

$$\displaystyle \int_{0}^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}$$ then $$\displaystyle \lambda $$ equals