Integrals Test ...

Let f be a function defined for every x, such that f'' = -f ,f(0)=0, f' (0) = 1, then f(x) is equal to

Let $$\displaystyle \frac{d}{dx}F(x)=\frac{e^{{s}{m}{x}}}{x}$$ , $$x>0$$. lf $$\displaystyle \int_{1}^{4}\frac{3}{x}e^{{s}{m}{x}^{3}}dx=F(k)-F(1)$$, then one of the possible values of $${k}$$ is

$$\displaystyle \int_{0}^{1}\displaystyle \frac{x(2x^{2}+1)}{x^{8}+2x^{6}-x^{2}+1}dx=$$

If $$\mathrm{a}>\mathrm{b}$$ then $$\displaystyle \int_{0}^{\pi}\frac{\mathrm{d}\mathrm{x}}{\mathrm{a}+\mathrm{b}\sin \mathrm{x}}=$$

If $$I_{n}=\displaystyle \int_{0}^{\infty}e^{-x}x^{n-1} dx$$, then $$\displaystyle \int_{0}^{\infty}e^{-\lambda x}x^{n-1} dx$$ is equal to?

The value of $$\displaystyle \int_{0}^{\pi}\dfrac {dx}{1+2sin^2x}$$ is

If $$f(x) = x - x^2 +1$$ & $$g(x)=max\left \{ f(t) ;0\leq t< x \right \}$$, then $$\overset {1}{\underset { 0 }{ \int } }  g (x) dx = ?$$

What is the value of a + b ?

The value of $$\displaystyle {\int}_0^{\pi}\frac{dx}{1+2 \sin^2 x}$$ is

$$\displaystyle \int_{- \dfrac{\pi}{2}}^{\dfrac{\pi}{2}}\sin^{2}x. \cos^{3} x dx$$ is equal to