Fundamental Theorem of Integral Calculus

Q. The integral $\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-x^2/2}dx$ is equal to

1. $\frac{1}{2}$
2. $\frac{1}{\sqrt{2}}$
3. $1$
4. $\alpha$

Q. The values of integrals ${\int }_0^1\left({\int }_0^1\frac{x-y}{(x+y{)}^3}dy\right)dx$ and ${\int }_0^1\left({\int }_0^1\frac{x-y}{(x+y{)}^3}dx\right)dy$ are

1. same and equal to $0.5$
2. same and equal to $-0.5$
3. $0.5$ and $-0.5,$ respectively
4. $-0.5$ and $0.5,$ respectively

Q. The value of the line integral $\frac{2}{\pi }{\oint }_c(-y^{3}dx+x^{3}dy)$,
where C is the circles $x^2+y^2=1$ oriented counter - clockwise, is

1. 0
2. 1
3. 3
4. 6

Q.

Let $f:R\to R$ be a continuous function defined by $f\left(x\right)=\frac{1}{e^x+2e^{-x}}$

Statement I : $f\left(c\right)=\frac{1}{3}$, for some $c\in R$.

Statement II : $0<f\left(x\right)\le \frac{1}{2\sqrt{2}},\forall x\in R$

1. Statement I is correct, Statement II is also correct; Statement II is the correct explanation of Statement I

2. Statement I is correct, Statement II is also correct; Statement II is not the correct equation of Statement I

3. Statement I is correct, statement II is correct

4. Statement I is incorrect, Statement II is correct

Q. The value of the integral ${\int }_0^{2\pi }\left(\frac{3}{9+si{n}^2\theta }\right)d\theta$ is

1. $\frac{2\pi }{\sqrt{10}}$
2. $2\sqrt{10}\pi$
3. $\sqrt{10}\pi$
4. $2\pi$

Q. The value of the following definite integral is (round off to three decimal places).
${\int }_1^e\left(xlnx\right)dx$

1. 2.09

Q. Which of the following integrals is unbounded?

1. ${\int }_0^{\pi /4}tanxdx$
2. ${\int }_0^{\infty }\frac{1}{x^2+1}dx$
3. ${\int }_0^{\infty }x{e}^{-x}dx$
4. ${\int }_0^1\frac{1}{1-x}dx$

Q.

${\int }_{-2}^2\left(x-\left|x\right|\right)dx=$

1. $0$

2. $2$

3. $4$

4. $-4$

Q. If $f\left(x\right)=Rsin\left(\frac{\pi x}{2}\right)+S\&f^{\prime }\left(\frac{1}{2}\right)=\sqrt{2}$ and ${\int }_0^{1}f\left(x\right)dx=\frac{2R}{\pi },$ then the constants $R$ and $S$ are, respectively.

1. $\frac{2}{\pi }$ and $\frac{16}{\pi }$
2. $\frac{2}{\pi }$ and $0$
3. $\frac{4}{\pi }$ and $0$
4. $\frac{4}{\pi }$ and $\frac{16}{\pi }$

Q. The value of the integral ${\int }_0^{\pi }xco{s}^{2}xdx$ is

1. ${\pi }^2/8$
2. ${\pi }^2/4$
3. ${\pi }^2/2$
4. ${\pi }^2$