Integrals Test ...

 If $$ b > a$$ and $$ \displaystyle I = \int_{a}^{b}\frac{dx}{\sqrt{ (x-a)(b-x)}}$$ then $$I$$ equals

A function $$f $$ is defined by $$\displaystyle f(x)=\frac{1}{2^{r-1}},\frac{1}{2r}<x\leq \frac{1}{2^{r-1}},r=1,2,3,.....$$ then the value of $$ \displaystyle \int _{0}^{1}f(x)dx $$ is equal

Evaluate $$\displaystyle \int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt{x+x^{2}+x^{3}}}$$

$$\displaystyle \int_{e^{e^{e}}}^{e^{e^{e^{e}}}}\frac{dx}{xlnx\cdot ln\left ( lnx \right )\cdot ln\left ( ln\left ( lnx \right ) \right )}$$ equals

The value of integral $$\displaystyle \int _{ \frac { \pi  }{ 6 }  }^{ \frac { \pi  }{ 3 }  }{ \cfrac { \sin { x } -x\cos { x }  }{ x(x+\sin { x } ) }  } dx$$ is

Evaluate $$\displaystyle \int_{0}^{1}\left ( tx+1-x \right )^{n}dx,$$ where n is a positive integer and t is a parameter independent of x. Hence $$\displaystyle \int_{0}^{1}x^{k}\left ( 1-x \right )^{n-k}dx=\frac{P}{\left [ ^{n}C_{k}\left ( n+1 \right ) \right ]}for\:k=0,1,......n$$, then $$P=$$

Evaluate: $$\displaystyle \int_{0}^{1}\dfrac{1-x}{1+x}\cdot \dfrac{dx}{\sqrt{x+x^{2}+x^{3}}}$$

Evaluate $$\displaystyle \int_{0}^{\pi /4}\frac{\cos x-\sin x}{10+\sin 2x}dx$$

The value of $$\displaystyle \int_{3}^{4}\sqrt {(4 - x)(x - 3)}dx$$ is

The value of the integral $$\displaystyle \int_{\frac {1}{3}}^1\frac {(x-x^3)^{\frac {1}{3}}}{x^4}dx$$