If $f(x)$ is a periodic function of period $T$; then

$\underset{0}{\overset{\mathrm{nT}}{\int }}f(\mathrm{x})\mathrm{dx}=\mathrm{n}\underset{0}{\overset{\mathrm{T}}{\int }}f(\mathrm{x})\mathrm{d}\mathrm{x}$

Solved Examples

1. $\underset{0}{\overset{10}{\int }}{\mathrm{e}}^{\{x\}}\mathrm{dx}=$

(a) $10(\mathrm{e}-1)$

(b) 10

2. $\underset{-\pi /2}{\overset{199\pi /2}{\int }}\sqrt{1+\cos 2\mathrm{x}}\mathrm{dx}$ is equal to

(a) $50\sqrt{2}$

(b) $100\sqrt{2}$

(c) $150\sqrt{2}$

(d) $200\sqrt{2}$

3. Let $f$ be a real valued function satisfying $f(\mathrm{x})+f(\mathrm{x}+6)=f(\mathrm{x}+3)+f(\mathrm{x}+9)$. Then $\underset{\mathrm{x}}{\overset{\mathrm{x}+12}{\int }}f(\mathrm{t})\mathrm{dt}$ is

(a) linear function of $\mathrm{x}$

(b) exponential function of $\mathrm{x}$

(c) a constant

(d) None of these

4. The value of $\underset{-\pi /4}{\overset{n\pi -\pi /4}{\int }}|\sin x+\cos x|dx$ is

(a) $\sqrt{2}\mathrm{n}$

(B) $2\sqrt{2}\mathrm{n}$

(c) $4\sqrt{2}\mathrm{n}$

(d) None of these

5. If $\underset{a}{\overset{b}{\int }}|\sin x|dx=8$ and $\underset{0}{\overset{a+b}{\int }}|\cos x|dx=9$

then the value of $a$ and $b$ are

(a) ${\displaystyle \frac{\pi }{4}},{\displaystyle \frac{17\pi }{4}}$

(b) ${\displaystyle \frac{3\pi }{4}},{\displaystyle \frac{19\pi }{4}}$

(c) ${\displaystyle \frac{\pi }{6}},{\displaystyle \frac{25\pi }{6}}$

(d) None of these

6. If $\mathrm{A}=\underset{1}{\overset{\sin \theta }{\int }}{\displaystyle \frac{\mathrm{tdt}}{1+{\mathrm{t}}^2}}$ & $\mathrm{B}=\underset{1}{\overset{\mathrm{cosec}\theta }{\int }}{\displaystyle \frac{\mathrm{dt}}{\mathrm{t}(1+{\mathrm{t}}^2)}}$, then the value of $\left|\begin{array}{ccc}\mathrm{A}& {\mathrm{A}}^2& \mathrm{B}\\ {\mathrm{e}}^{\mathrm{A}}{\mathrm{e}}^{\mathrm{B}}& {\mathrm{B}}^2& -1\\ 1& {\mathrm{A}}^2+{\mathrm{B}}^2& -1\end{array}\right|$ is

(a) $\sin \theta$

(b) $\mathrm{cosec}\theta$

(c) 0

(d) 1

7. $\underset{e^{-1}}{\overset{\tan x}{\int }}{\displaystyle \frac{\mathrm{t}d\mathrm{t}}{1+{\mathrm{t}}^2}}+\underset{{\mathrm{e}}^{-1}}{\overset{\cos \mathrm{x}}{\int }}{\displaystyle \frac{\mathrm{dt}}{\mathrm{t}(1+{\mathrm{t}}^2)}}=$

(a) 1

(b) e

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

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

Exercise

1. $\mathrm{I}=\underset{0}{\overset{10}{\int }}\sin \mathrm{x}\mathrm{dx}$

(a) 10

(b) $10(1-\cos 1)$

(c) $5(1-\cos 1)$

(d) $5\cos 1$

2. $\underset{\pi }{\overset{10\pi }{\int }}|\sin x|dx=$

(a) 20

(b) 8

(c) 10

(d) 18

3. If ${\mathrm{I}}_1=\underset{0}{\overset{3\pi }{\int }}f({\cos }^2\mathrm{x})\mathrm{dx}$ & ${\mathrm{I}}_2=\underset{0}{\overset{\pi }{\int }}f({\cos }^2\mathrm{x})\mathrm{dx}$ then

(a) ${\mathrm{I}}_1={\mathrm{I}}_2$

(b) ${\mathrm{I}}_1=2{\mathrm{I}}_2$

(c) ${\mathrm{I}}_1=5{\mathrm{I}}_2$

(d) ${\mathrm{I}}_1=3{\mathrm{I}}_2$

4. $\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{(n!{)}^{{\displaystyle \frac{1}{n}}}}{n^n}}$ is equal to

(a) $\mathrm{e}$

(b) ${\displaystyle \frac{1}{\mathrm{e}}}$

(c) e-1

(d) $\mathrm{e}-5$

5. The value of the definite integral $\underset{0}{\overset{1}{\int }}{\displaystyle \frac{\mathrm{xdx}}{{\mathrm{x}}^3+16}}$ lies in the interval $[\mathrm{a},\mathrm{b}]$. Then smallest such interval is

(a) $[0,{\displaystyle \frac{1}{17}}]$

(b) $[0,1]$

(c) $[0,{\displaystyle \frac{1}{27}}]$

(d) $[{\displaystyle \frac{1}{17}},1]$

6. Let $f:\mathrm{R}\to \mathrm{R}$ such that $f(\mathrm{x}+2\mathrm{y})=f(\mathrm{x})+f(2\mathrm{y})+4\mathrm{xy},\mathrm{\forall }\mathrm{x},\mathrm{y}\in \mathrm{R}$ & $f(0)=0$ If ${\mathrm{I}}_1=\underset{0}{\overset{1}{\int }}f(\mathrm{x})\mathrm{dx}$,

${\mathrm{I}}_2=\underset{0}{\overset{1}{\int }}f(\mathrm{x})\mathrm{dx}$ & ${\mathrm{I}}_3=\underset{{\displaystyle \frac{1}{2}}}{\overset{2}{\int }}f(\mathrm{x})\mathrm{dx}$, then

(a) ${\mathrm{I}}_1={\mathrm{I}}_2>{\mathrm{I}}_3$

(b) ${\mathrm{I}}_1>{\mathrm{I}}_2>{\mathrm{I}}_3$

(c) ${\mathrm{I}}_1=$ ${\mathrm{I}}_2<{\mathrm{I}}_3$

(d) ${\mathrm{I}}_1<$ ${\mathrm{I}}_2<$ ${\mathrm{I}}_3$

7. The integral $\underset{{\tan }^{-1}\lambda }{\overset{{\cot }^{-1}\lambda }{\int }}{\displaystyle \frac{\tan x}{\tan x+\cot x}}dx,\mathrm{\forall }\lambda \in \mathrm{R}$ can not take the value

(a) ${\displaystyle \frac{-\pi }{4}}$

(b) ${\displaystyle \frac{-\pi }{2}}$

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

(d) ${\displaystyle \frac{3\pi }{4}}$

8. Read the following and answer the questions that follow :-

$I=\underset{0}{\overset{10\pi }{\int }}{\displaystyle \frac{\cos 6x\cos 7x\cos 8x\cos 9x}{1+e^{2{\sin }^{3}4x}}}dx$

(i) If $I=k\int limit{s}_0^{{\displaystyle \frac{\pi }{2}}}\cos 6\mathrm{x}\cos 7\mathrm{x}\cos 8\mathrm{x}\cos 9\mathrm{xdx}$, then $\mathrm{k}=$

(a) 5

(b) 10

(c) 20

(d) None

(ii) If $I=c\underset{0}{\overset{{\displaystyle \frac{\pi }{2}}}{\int }}\cos 6x\cos 8x\cos 2xdx$, then $c=$

(a) 5

(b) 10

(c) 20

(d) None

(iii) The value of $\mathrm{I}=$

(a) ${\displaystyle \frac{5\pi }{4}}$

(b) ${\displaystyle \frac{5\pi }{8}}$

(c) ${\displaystyle \frac{5\pi }{16}}$

(d) ${\displaystyle \frac{5\pi }{12}}$

9. Match the following if

$I_1=\underset{0}{\overset{1}{\int }}{\displaystyle \frac{e^{{\tan }^{-1}x}}{\sqrt{1+{\mathrm{x}}^2}}}dx,I_2=\underset{0}{\overset{1}{\int }}{\displaystyle \frac{x^{{\tan }^{-1}x}}{\sqrt{1+{\mathrm{x}}^2}}}dx,I_3=\underset{0}{\overset{1}{\int }}{\displaystyle \frac{x^2{e}^{{\tan }^{-1}x}}{\sqrt{1+{\mathrm{x}}^2}}}dx$, then

10. Let $S_n=\sum _{k=0}^n{\displaystyle \frac{n}{n^2+kn+k^2}}$ & $T_n=\sum _{k=0}^{n-1}{\displaystyle \frac{n}{n^2+kn+k^2}}$ for $n=1,2,3$ then

(a) ${\mathrm{S}}_{\mathrm{n}}<{\displaystyle \frac{\pi }{3\sqrt{3}}}$

(b) ${\mathrm{S}}_{\mathrm{n}}>{\displaystyle \frac{\pi }{3\sqrt{3}}}$

(c) ${\mathrm{T}}_{\mathrm{n}}<{\displaystyle \frac{\pi }{3\sqrt{3}}}$

(d) ${\mathrm{T}}_{\mathrm{n}}>{\displaystyle \frac{\pi }{3\sqrt{3}}}$

11. $\underset{0}{\overset{[x]}{\int }}{\displaystyle \frac{2^x}{2^{[x]}}}dx$ is

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

(b) ${\displaystyle \frac{[\mathrm{x}]}{{\log }_{\mathrm{e}}2}}$

(c) ${\displaystyle \frac{1}{2}}{\displaystyle \frac{[\mathrm{x}]}{{\log }_{\mathrm{e}}2}}$

(d) None of these.

12. The number of positive continuous functions $f(\mathrm{x})$ defined in $[0,1]$ for which $\underset{0}{\overset{1}{\int }}f(\mathrm{x})\mathrm{dx}=1$,

$\underset{0}{\overset{1}{\int }}\mathrm{x}f(\mathrm{x})\mathrm{dx}=\mathrm{a},\underset{0}{\overset{1}{\int }}{\mathrm{x}}^{2}f(\mathrm{x})\mathrm{d}\mathrm{x}={\mathrm{a}}^2$ is

(a) one

(b) infinite

(c) two

(d) zero