Integrals Test ...

The value of $$\int_{a}^{b}(x-a)^{3}(b-x)^{4} d x$$ is

Given $$I_{m}=\displaystyle \int_{1}^{e}(\log x)^{m} d x .$$ If $$\dfrac{I_{m}}{K}+\dfrac{I_{m-2}}{L}=e,$$ then the values of $$K$$ and $$L$$ are

If $$\int_{0}^{f(x)} t^{2} d t=x \cos \pi x,$$ then $$f^{\prime}({9})$$ is

The value of the definite integral $$\int_{2}^{4}(x(3-x)(4+x)(6-x)$$ $$(10-x)+\sin x) d x$$ equals

Given, $$f(x) = \begin{vmatrix} 0 & {x^2 - \sin x} & {\cos x - 2} \\ {\sin x -x^2} & 0 & {1 - 2x} \\ {2 - \cos x} & {2x - 1} & 0 \end{vmatrix}$$, then $$\displaystyle \int f(x) dx$$ is equal to

Let $$f(x)=\displaystyle \int_{-1}^{x}e^{t^2}dt$$ and $$h(x)=f(1+g(x))$$ where $$g(x)$$ is defined for all $$x, g'(x)$$ exists for all $$x,$$ and $$g(x) < 0$$ for $$x>0.$$ If $$h'(1)=e$$ and $$g'(1)=1,$$ then the possible values which $$g(1)$$ can take 

Let $$f : R \rightarrow R$$ be a function as $$f(x) = (x - 1)(x + 2)(x - 3)(x - 6) - 100$$. If $$g(x)$$ is a polynomial of degree $$\leq 3$$ such that $$\displaystyle \int \frac{g(x)}{f(x)} dx$$ does not contain any logarithm function and $$g(-2) = 10$$. Then

$$\displaystyle \int \frac{g(x)}{f(x)} dx$$, equals

If $$\displaystyle \int_{-2}^{-1} (ax^2-5)dx $$ and $$5+\displaystyle \int_{1}^{2} (bx+c)dx=0, $$ then 

The value of the definite integral 
$$\displaystyle \int_{0}^{\infty} \dfrac{dx}{(1+x^a)(1+x^2)}(a>0)$$ is