If $f(\mathrm{x})$ is continuous on $[\mathrm{a},\mathrm{b}]$ and $\mathrm{F}$ is an antiderivative (primitive) of $f$.

i.e. $f(\mathrm{x})={\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}\mathrm{F}$, then

${{\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}=\mathrm{F}(\mathrm{x})]}_{\mathrm{a}}^{\mathrm{b}}=\mathrm{F}(\mathrm{b})-\mathrm{F}(\mathrm{a})$

Note: ${\int }_a^{a}f(x)dx=0$

Geometrical interpretation of definite integral

If $f(\mathrm{x})>0\mathrm{\forall }\mathrm{x}\in [\mathrm{a},\mathrm{b}]$, then ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}$ is numerically equal to the area bonded by the curve $\mathrm{y}=f(\mathrm{x})$, the $\mathrm{x}$ - axis and the straight lines $\mathrm{x}$ = $\mathrm{a}$ & $\mathrm{x}$ = $\mathrm{b}$.

Properties of definite integrals

1. ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{t})\mathrm{dt}$

2. ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{d}\mathrm{x}=-{\int }_{\mathrm{b}}^{\mathrm{a}}f(\mathrm{x})\mathrm{d}\mathrm{x}$

3. ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{c}}f(\mathrm{x})\mathrm{dx}+{\int }_{\mathrm{c}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}$ where $\mathrm{c}\in \mathrm{R}$

4. ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{a}-\mathrm{x})\mathrm{dx}$

5. ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{a}+\mathrm{b}-\mathrm{x})\mathrm{dx}$

6. ${\int }_{-a}^{a}f(\mathrm{x})\mathrm{d}x={\int }_0^{\mathrm{a}}(f(\mathrm{x})+f(-\mathrm{x}))\mathrm{d}\mathrm{x}$ = $\{\begin{array}{l}l2{\int }_0^{a}f(\mathrm{x})\mathrm{d}x\text{ if }f(-\mathrm{x})=f(\mathrm{x})\\ \text{ ie. if }f(\mathrm{x})\text{ is even }\\ 0,\text{ if }f(-\mathrm{x})=-f(\mathrm{x})\\ (\text{ ie. if }f(\mathrm{x})\text{ is odd })\end{array}$.

7. ${\int }_{\mathrm{a}}^{2\mathrm{a}}f(\mathrm{x})\mathrm{dx}={\int }_0^{\mathrm{a}}(f(\mathrm{x})+f(2\mathrm{a}-\mathrm{x}))\mathrm{dx}$ = $\{\begin{array}{l}2{\int }_0^{\mathrm{a}}f(\mathrm{x})\mathrm{dx}\text{ if }f(2\mathrm{a}-\mathrm{x})=f(\mathrm{x})\\ 0\text{ ,if }f(2\mathrm{a}-\mathrm{x})=-f(\mathrm{x})\end{array}$.

8. ${\int }_b^{a}f(x)dx=(b-a){\int }_0^{1}f((b-a)x+a)dx$

9. If $\mathrm{m}$ & $\mathrm{M}$ be the least and greatest values of $f(\mathrm{x})$ respectively on $[\mathrm{a},\mathrm{b}]$, then $\mathrm{m}(\mathrm{b}-\mathrm{a})\le {\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}\le \mathrm{M}$.

10. ${\int }_0^1{\mathrm{x}}^m(1-\mathrm{x}{)}^n\mathrm{dx}={\displaystyle \frac{m!n!}{(m+n+1)!}}$

11. ${\int }_{\mathrm{a}}^{\mathrm{b}}(f\circ g)(\mathrm{x}){\mathrm{g}}^1(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{g}(\mathrm{x})){\mathrm{g}}^1(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{g}(\mathrm{a})}^{\mathrm{g}(\mathrm{b})}f(\mathrm{t})\mathrm{dt}$

12. If $f(\mathrm{x})$ is a periodic function with period $\mathrm{T}$ then

i. ${\int }_0^{\mathrm{nT}}f(\mathrm{x})\mathrm{dx}=\mathrm{n}{\int }_0^{\mathrm{T}}f(\mathrm{x})\mathrm{dx}\mathrm{n}\in \mathrm{Z}$

ii. ${\int }_a^{a+nT}f(x)dx=n{\int }_a^{a+T}f(x)dx=n{\int }_0^{T}f(x)dx$

which is independent of a

iii. ${\int }_{a+n\mathrm{T}}^{b+n\mathrm{T}}f(\mathrm{x})\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx},\mathrm{n}\in \mathrm{Z}$

iv. ${\int }_{\mathrm{mT}}^{\mathrm{nT}}f(\mathrm{x})\mathrm{dx}=(\mathrm{n}-\mathrm{m}){\int }_0^{\mathrm{T}}f(\mathrm{x})\mathrm{dx},\mathrm{m},\mathrm{n}\in \mathrm{Z}$.

v. If $f(\mathrm{t})$ is an odd (even) function periodic with period $\mathrm{T}$, then $\mathrm{F}(\mathrm{x})={\int }_{\mathrm{a}}^{\mathrm{x}}f(\mathrm{t})\mathrm{dt}$ is an even (odd) function with period $\mathrm{T}$.

13. If $f(\mathrm{x})\le \mathrm{g}(\mathrm{x})$ on $[\mathrm{a},\mathrm{b}]$, then ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}\le {\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{g}(\mathrm{x})\mathrm{dx}$.

Also ${\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{d}\mathrm{x}\le {\int }_{\mathrm{a}}^{\mathrm{b}}|f(\mathrm{x})|\mathrm{d}\mathrm{x}$

14. Leibnitz Rule

i. ${\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}{\int }_{\phi (\mathrm{x})}^{\mathrm{\Psi }(\mathrm{x})}f(\mathrm{t})\mathrm{dt}=f(\mathrm{\Psi }(\mathrm{x})){\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}(\mathrm{\Psi }(\mathrm{x}))-f(\phi (\mathrm{x})){\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}(\phi (\mathrm{x}))$

ii. ${\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}{\int }_{\phi (\mathrm{x})}^{\mathrm{\Psi }(\mathrm{x})}f(\mathrm{x},\mathrm{t})\mathrm{dt}={\int }_{\phi (\mathrm{x})}^{\mathrm{\Psi }(\mathrm{x})}{\displaystyle \frac{\partial f(\mathrm{x},\mathrm{t})}{\partial \mathrm{x}}}\mathrm{dt}+{\displaystyle \frac{\mathrm{d}\mathrm{\Psi }(\mathrm{x})}{\mathrm{dx}}}f(\mathrm{x},\mathrm{\Psi }(\mathrm{x}))-{\displaystyle \frac{{\mathrm{d}}_{\phi }(\mathrm{x})}{\mathrm{dx}}}f(\mathrm{x},\phi (\mathrm{x}))$

Reduction Formulae

${\mathrm{I}}_{\mathrm{n}}={\int }_0^{\pi /2}{\sin }^{\mathrm{n}}\mathrm{xdx}={\int }_0^{\pi /2}{\cos }^{\mathrm{n}}\mathrm{xdx}={\displaystyle \frac{\mathrm{n}-1}{\mathrm{n}}}{\mathrm{I}}_{\mathrm{n}-2},\mathrm{n}\in {\mathrm{Z}}^{+}$

1. = $\{\begin{array}{l}\left({\displaystyle \frac{n-1}{n}}\right)\left({\displaystyle \frac{n-3}{n-2}}\right)\dots \dots ..{\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{\pi }{2}},\text{ If }n\text{ is even }\\ & \\ \left({\displaystyle \frac{n-1}{n}}\right)\left({\displaystyle \frac{n-3}{n-2}}\right)\dots \dots \dots {\displaystyle \frac{2}{3}},\text{ If }n\text{ is odd. }\end{array}$.

2. ${\mathrm{I}}_{\mathrm{n}}={\int }_0^{\pi /4}{\tan }^{\mathrm{n}}\mathrm{x}d\mathrm{x}\mathrm{n}>1$,

${\mathrm{I}}_{\mathrm{n}}+{\mathrm{I}}_{\mathrm{n}-2}={\displaystyle \frac{1}{\mathrm{n}-1}}$

$I_n={\displaystyle \frac{1}{n-1}}-{\displaystyle \frac{1}{n-3}}+{\displaystyle \frac{1}{n-5}}-{\displaystyle \frac{1}{n-7}}\dots \dots \dots \dots \dots I_1\text{ or }{I}_0$

according as $\mathrm{n}$ is odd or even.

$({\mathrm{I}}_0={\displaystyle \frac{\pi }{4}},\mathrm{I},={\displaystyle \frac{1}{2}}{\log }_{\mathrm{e}}2={\log }_{\mathrm{e}}\sqrt{2})$

iii. ${\int }_0^{\pi /4}({\tan }^{n}x+{\tan }^{n-2}x)dx={\int }_{\pi /4}^{\pi /2}({\cot }^{n}x+{\cot }^{n-2}x)dx={\displaystyle \frac{1}{n-1}}$

Walli’s Formulae.

$\begin{array}{rl}& I_{m,n}={\int }_0^{\pi /2}({\sin }^{m}x{\cos }^{n}x)dx\\ \\ & ={\displaystyle \frac{m-1}{m+1}}{I}_{m-2,n}\\ \\ & ={\displaystyle \frac{n-1}{m+1}}{I}_{m,n-2}\text{ where }m,n\in Z^x\\ \\ & I_{m,n}={\int }_0^{\pi /2}({\sin }^{m}x{\cos }^{n}x)dx\\ \\ & ={\displaystyle \frac{((m-1)(m-3)\dots \dots .1\text{ or }2)((n-1)(n-3)\dots \dots ..1\text{ or }2)}{(m+n)(m+n-2)(m+n-4)\dots \dots ..(1\text{ or }2)}}k\end{array}$

where $\mathrm{k}={\displaystyle \frac{\pi }{2}}$ if both $\mathrm{m}$ & $\mathrm{n}$ are even

$\mathrm{k}=1\text{ otherewise. }$

Improper integral

If $f(\mathrm{x})$ is continuous on $[\mathrm{a},\mathrm{\infty })$ then ${\int }_{\mathrm{a}}^{\mathrm{\infty }}f(\mathrm{x})\mathrm{dx}=\underset{\mathrm{b}\to \mathrm{\infty }}{lim}{\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}$ is called an improper integral.

If there exists a finite limit on right side of the above equation, we say the improper integral is convergent, otherwise it is divergent.

Similarity ${\int }_{-\mathrm{\infty }}^{b}f(x)dx=\underset{\mathrm{a}\to -\mathrm{\infty }}{lim}{\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}$

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\mathrm{x})\mathrm{dx}={\int }_{-\mathrm{\infty }}^{\mathrm{a}}f(\mathrm{x})\mathrm{dx}+{\int }_{\mathrm{a}}^{\mathrm{\infty }}f(\mathrm{x})\mathrm{dx}$

Geometrically, for $f(\mathrm{x})>0,{\int }_{\mathrm{a}}^{\mathrm{\infty }}f(\mathrm{x})\mathrm{dx}$ gives area of the figure bounded by $\mathrm{y}=f(\mathrm{x}),\mathrm{x}-$ axis & $\mathrm{x}=\mathrm{a}$.

Gamma Function

If $n$ is a positive rational number, then the improper integral ${\int }_0^{\mathrm{\infty }}{e}^{-\mathrm{x}}{\mathrm{x}}^{n-1}\mathrm{dx},\mathrm{x}\in Q^{+}$ is defined as

Gamma function and is denoted by $\cdot \mathrm{\Gamma }\mathrm{n}$

$\therefore \mathrm{\Gamma }\mathrm{n}={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{x}}{\mathrm{x}}^{\mathrm{n}-1}\mathrm{dx},\mathrm{x}\in {\mathrm{Q}}^{+}$

Properties

i. $\mathrm{\Gamma }\mathrm{n}+1=\mathrm{n}\mathrm{\Gamma }\mathrm{n}=\mathrm{n}!;\mathrm{n}\in \mathrm{N}$

ii. $\mathrm{\Gamma }1=1$ and $\mathrm{\Gamma }0=\mathrm{\infty },{\mathrm{\Gamma }}_{{\displaystyle \frac{1}{2}}}=\sqrt{\pi }$

iii. ${\int }_0^{\pi /2}{\sin }^{\mathrm{m}}x{\cos }^{\mathrm{n}}\mathrm{xdx}={\displaystyle \frac{{\displaystyle \frac{{\mathrm{\Gamma }}_{\mathrm{m}+1}}{2}}{\displaystyle \frac{{\mathrm{\Gamma }}_{\mathrm{n}+1}^2}{2}}}{2{\displaystyle \frac{\sqrt[\mathrm{m}+\mathrm{n}+2]{2}}{2}}}}$

iv. $\mathrm{\Gamma }\mathrm{n}\mathrm{\Gamma }\mathrm{n}-1$ = ${\displaystyle \frac{\pi }{\sin \mathrm{n}\pi }},0<\mathrm{n}<1$.

v. $\mathrm{\Gamma }\mathrm{n}{\mathrm{\Gamma }}_{\mathrm{m}+{\displaystyle \frac{1}{2}}}={\displaystyle \frac{\sqrt{\pi }}{2^{2\mathrm{m}-1}}}\mathrm{\Gamma }2\mathrm{m}$

vi. ${\mathrm{\Gamma }}_{{\displaystyle \frac{1}{n}}}{\mathrm{\Gamma }}_{{\displaystyle \frac{2}{n}}}{\mathrm{\Gamma }}_{{\displaystyle \frac{3}{n}}}\cdots ..{\mathrm{\Gamma }}_{{\displaystyle \frac{n-1}{n}}}={\displaystyle \frac{(2\pi {)}^{{\displaystyle \frac{n-1}{2}}}}{n^{{\displaystyle \frac{1}{2}}}}}$

Beta Function

The Beta function is denoted by $B(m,n)$ where $B(m,n)={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx,m,n>0$

Properties

$\begin{array}{lll}\text{i. }& B(m,n)& ={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx,m,n>0\\ \\ & & ={\int }_0^1{\displaystyle \frac{x^{m-1}}{(1+x{)}^{m+n}}}dx\\ \\ & & ={\displaystyle \frac{\mathrm{\Gamma }\mathrm{m}\mathrm{\Gamma }\mathrm{n}}{\mathrm{\Gamma }(\mathrm{m}+\mathrm{n})}}\end{array}$

Integration as limit of a sum

${\int }_0^{1}f(\mathrm{x})\mathrm{dx}=\underset{\mathrm{n}\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{r}=1}^{\mathrm{n}}f\left({\displaystyle \frac{\mathrm{r}-1}{\mathrm{n}}}\right)=\underset{\mathrm{n}\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{r}=0}^{\mathrm{n}-1}f\left({\displaystyle \frac{\mathrm{r}}{\mathrm{n}}}\right)$

Here replace ${\displaystyle \frac{\mathrm{r}}{\mathrm{n}}}$ by $x$

replace ${\displaystyle \frac{1}{\mathrm{n}}}$ by $\mathrm{dx}$

and $\underset{n\to \mathrm{\infty }}{lim}\sum$ by $\int$

Solved Examples

1. If ${\int }_0^1{\displaystyle \frac{e^{t}dt}{t+1}}=a$, then ${\int }_{b-1}^b{\displaystyle \frac{e^{-t}dt}{t-b-1}}$ is equal to

(a) ${\mathrm{ae}}^{-\mathrm{b}}$

(b) $-{\mathrm{ae}}^{-\mathrm{b}}$

(c) $-b{e}^{-a}$

(d) $a{e}^b$

2. If ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{\mathrm{x}}^2}\mathrm{dx}={\displaystyle \frac{\sqrt{\pi }}{2}}$, then ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{ax}}\mathrm{dx}(\mathrm{a}>0)$ is equal to

(a) ${\displaystyle \frac{\sqrt{\pi }}{2}}$

(b) ${\displaystyle \frac{\sqrt{\pi }}{2\mathrm{a}}}$

(c) ${\displaystyle \frac{2\sqrt{\pi }}{\mathrm{a}}}$

(d) ${\displaystyle \frac{\sqrt{\pi }}{2\sqrt{\mathrm{a}}}}$

3. Given ${\int }_1^2{\mathrm{e}}^{{\mathrm{x}}^2}\cdot \mathrm{dx}=\mathrm{a}$, then the value of ${\int }_{\mathrm{e}}^{{\mathrm{e}}^4}\sqrt{{\log }_{\mathrm{e}}\mathrm{x}}\mathrm{dx}$ is equal to

(a) ${\mathrm{e}}^4-\mathrm{e}$

(b) ${\mathrm{e}}^4-\mathrm{a}$

(c) $2{\mathrm{e}}^4-\mathrm{a}$

(d) $2e^4-e-a$

4. ${\int }_0^1{\displaystyle \frac{\sin t.dt}{t+1}}=\alpha$, then the value of

${\int }_{4\pi -2}^{4\pi }{\displaystyle \frac{\sin {\displaystyle \frac{\mathrm{t}}{2}}\mathrm{dt}}{4\pi +2-\mathrm{t}}}$ is equal to

(a) $\alpha$

(b) $-\alpha$

(c) $\pi \alpha$

(d) $2\alpha$

5. Let $f$ be a continuous function. Let

$\begin{array}{r}{I}_1={\int }_{{\sin }^{2}t}^{1+{\cos }^{2}t}xf\{x(2-x)\}dx\text{ and }{I}_2={\int }_{{\sin }^{2}t}^{1+{\cos }^{2}t}f\{x(2-x)\}dx\text{, then }{I}_1:I_2=\end{array}$

(a) 0

(b) 1

(c) 2

(d) none

6. If $f(\mathrm{x})=\int {\displaystyle \frac{{\log }_{\mathrm{e}}\mathrm{xdx}}{1+\mathrm{x}}}$, then $f(\mathrm{e})+f\left({\displaystyle \frac{1}{\mathrm{e}}}\right)$ is equal to

(a) 2

(b) ${\displaystyle \frac{1}{2}}$

(c) 1

(d) none

7. ${\int }_0^{\mathrm{x}}f(\mathrm{t})\mathrm{dt}=\mathrm{x}+{\int }_{\mathrm{x}}^1\mathrm{t}f(\mathrm{t})\mathrm{dt}$, then $f(1)$ is equal to

(a) ${\displaystyle \frac{1}{2}}$

(b) 0

(c) 1

(d) ${\displaystyle \frac{-1}{2}}$

8. $f(\mathrm{x})=\cos \mathrm{x}-{\int }_0^y(\mathrm{x}-\mathrm{t})f(\mathrm{t})\mathrm{dt}$, then $f^{\prime \prime }(\mathrm{x})+f(\mathrm{x})$ is equal to

(a) $-\cos x$

(b) $-\sin x$

(c) 0

(d) none of these

Exercise:

1. Let $\mathrm{I}={\int }_0^1{\displaystyle \frac{\sin \mathrm{x}}{\sqrt{\mathrm{x}}}}\mathrm{dx}$ and $\mathrm{J}={\int }_0^1{\displaystyle \frac{\cos \mathrm{x}}{\sqrt{\mathrm{x}}}}\mathrm{dx}$. Then which one of the following is true?

(a) $\mathrm{I}<{\displaystyle \frac{2}{3}}$ & $\mathrm{J}>2$

(b) $\mathrm{I}>{\displaystyle \frac{2}{3}}$ & $\mathrm{J}<2$

(c) $\mathrm{I}>{\displaystyle \frac{2}{3}}$ & $\mathrm{J}>2$

(d) $\mathrm{I}<{\displaystyle \frac{2}{3}}$ & $\mathrm{J}<2$

2. The equaiton of a curve is $\mathrm{y}=f(\mathrm{x})$. The tangent at $(1,f(1)),(2,f(2))$ & $(3,f(3))$ make angles ${\displaystyle \frac{\pi }{6}},{\displaystyle \frac{\pi }{3}}$ & ${\displaystyle \frac{\pi }{4}}$ respectively with the positive direction of the $\mathrm{x}-$ axis. Then the value of ${\int }_2^3{f}^{\prime }(\mathrm{x})f^{\prime \prime }(\mathrm{x})\mathrm{dx}+{\int }_1^3{f}^{\prime \prime }(\mathrm{x})\mathrm{dx}=$

(a) ${\displaystyle \frac{1}{\sqrt{3}}}$

(b) ${\displaystyle \frac{-1}{\sqrt{3}}}$

(c) 0

(d) none of these

3. $\underset{\lambda \to 0}{lim}{({\int }_0^1(1+x{)}^{\lambda }dx)}^{1/\lambda }$ is equal to

(a) $2{\log }_{\mathrm{e}}2$

(b) ${\displaystyle \frac{4}{\mathrm{e}}}$

(c) ${\log }_{\mathrm{e}}(4/\mathrm{e})$

(d) $4$

4. If $\beta +2{\int }_0^1{x}^2{e}^{-x^2}dx={\int }_0^1{e}^{-x^2}dx$, then the value of $\beta$ is

(a) $e^{-1}$

(b) e

(c) ${\displaystyle \frac{1}{2}}\mathrm{e}$

(d) cannot be determined

5.* $$ If $f(a)={\int }_{a-1}^{a+1}{\displaystyle \frac{dx}{1+x^8}}$, then the value of a for which $f(a)$ attains maximum is

(a) at $\mathrm{a}=0$

(b) at one value of a only

(c) at two values of a, one is $(-1,0)$ and the other is $(0,1)$

(d) at no value of a

6.* If $I_1={\int }_0^{\pi /4}(\tan x{)}^{\cot x}dx,I_2={\int }_0^{\pi /4}(\cot x{)}^{\tan x}dx,I_3={\int }_0^{\pi /4}(\tan x{)}^{\tan x}dx,I_4={\int }_0^{\pi /4}(\cot x{)}^{\cot x}dx$ then

(a) ${\mathrm{I}}_1<{\mathrm{I}}_3$

(b) ${\mathrm{I}}_2<{\mathrm{I}}_4$

(c) ${\mathrm{I}}_1<{\mathrm{I}}_4$

(d) ${\mathrm{I}}_3<{\mathrm{I}}_2$

7.* If $\mathrm{I}={\int }_0^1{\mathrm{e}}^{\sin x}\mathrm{dx}$, then

(a) $\mathrm{I}>0$

(b) $\mathrm{I}<\mathrm{e}$

(c) $\mathrm{I}>\mathrm{e}$

(d) $\mathrm{I}>{\mathrm{e}}^2$

8. If ${\int }_0^{10}f(x)dx=5$, then $\sum _{\mathrm{k}=1}^{10}{\int }_0^{1}f(\mathrm{k}-1+\mathrm{x})\mathrm{d}\mathrm{x}=$

(a) 50

(b) 20

(c) 5

(d) 0

9. The value of the definite integral ${\displaystyle \frac{{\int }_0^{\pi }(\sqrt[2013]{\cos x}+\sqrt[2013]{\sin x}+\sqrt[2013]{\tan x})dx}{{\int }_0^{\pi /2}\sqrt[2013]{\sin x}dx}}$ is

(a) 2

(b) 1

(c) ${\displaystyle \frac{1}{2}}$

(d) 0

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

(a) ${\displaystyle \frac{\pi }{2}}$

(b) 0

(c) $\pi$

(d) $2\pi$

11. The value of $f(x)={\int }_0^{{\sin }^{2}x}{\sin }^{-1}\sqrt{t}dt+{\int }_0^{{\cos }^{2}x}{\cos }^{-1}\sqrt{t}dt$ is

(a) ${\displaystyle \frac{\pi }{2}}$

(b) 1

(c) ${\displaystyle \frac{\pi }{4}}$

(d) none of these

12. The value of the definite integral ${\int }_0^1(1+{\mathrm{e}}^{-{\mathrm{x}}^2})\mathrm{dx}$ is

(a) -1

(b) 2

(c) $1+{\mathrm{e}}^{-1}$

(d) none of these