Property 2

Q. If ${\int }_{-a}^a(|x|+|x-2|)dx=22,(a>2)$ and $\left[x\right]$ denotes the greatest integer $\le x$, then ${\int }_a^{-a}(x+[x\left]\right)dx$ is equal to

Q. if a function f(x) is discontinuous in the interval (a, b) then ${\int }_a^{b}f\left(x\right)dx$ never exists.

1. False
2. True

Q. Calculate ${\int }_{-2}^{3}f\left(x\right)dx$ where
$f\left(x\right)=\{\begin{array}{cc}6& ifx>1\\ 3x^2& ifx\le 1\end{array}$

1. 19
2. 17
3. 15
4. 21

Q. The value of the definite integral ${\int }_{\frac{1}{b}}^b\frac{xlnx}{(a^2+x^2)(a^2{x}^2+1)}dx$ (where b > 0) equals

Q. The value of the integral $\underset{4}{\overset{10}{\int }}\frac{\left[x^2\right]}{[x^2-28x+196]+\left[x^2\right]}dx,$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is

Q.

Q. Solve ${\int }_0^{\infty }\frac{x{\tan }^{-1}x}{(1+x^2{)}^2}dx$

1. $\pi /2$
2. $\pi /6$
3. $\pi /4$
4. $\pi /8$

Q. Evaluate:${\int }_0^{\frac{\pi }{2}}{\cos }^{2}xdx$

Q. ${\int }_1^2(\frac{1}{x}-\frac{1}{2x^2})e^{2x}dx$.

Q. Solve : $\underset{0}{\stackrel{\infty }{\int }}\frac{x\ln x}{(1+x^2{)}^2}$