M L Aggarwal Solutions for Chapter: Integrals, Exercise 19: EXERCISE

Evaluate the integral $\int \frac{\log \left|x\right|}{(x+1{)}^3}dx$.

Evaluate the integral $\int e^x\left(\log x+\frac{1}{x^2}\right) dx$.

Evaluate the integral $\int \cos 2xl\mathrm{og}\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$.

Evaluate the integral $\int \frac{x^2-x+1}{{\left(x^2+1\right)}^{3/2}}{e}^{x}dx$.

Evaluate the integral $\int \frac{x^2-1}{\left(x^4+3x^2+1\right){\tan }^{-1}\left(x+{\displaystyle \frac{1}{x}}\right)}dx$.

Prove that ${\int }_0^{\mathrm{\pi }}\frac{x}{1-\cos a\sin x}dx=\frac{\mathrm{\pi }(\mathrm{\pi }-\mathrm{a})}{\sin a}$

Prove that ${\int }_0^{\mathrm{\pi }/2}\frac{f\left(\sin x\right)}{f\left(\sin x\right)+f\left(\cos x\right)}dx=\frac{\mathrm{\pi }}{4}$.

If p (x) is a polynomial of least degree that has maximum value equal to $6$ at $x=1$, and a minimum value equal to $2$ at $x=3$, then show that ${\int }_0^1 p\left(x\right) dx = \frac{19}{4}$.