Integral Calculus Definite Integrals(Lecture-03)

Property:

Limit as a sum

• Express the given series in the form of $\sum \dfrac{1}{\mathrm{n}} f\left(\dfrac{\mathrm{r}}{\mathrm{n}}\right)$

• Then the limit as its sum when $\mathrm{n} \rightarrow \infty$ i.e. $\lim \limits _{\mathrm{n} \rightarrow \infty} \sum \dfrac{1}{\mathrm{n}} . f\left(\dfrac{\mathrm{r}}{\mathrm{n}}\right)$

• Replace $\dfrac{\mathrm{r}}{\mathrm{n}}$ by $x$ and $\dfrac{1}{\mathrm{n}}$ by $\mathrm{dx}$ and $\dfrac{1}{\mathrm{n}}$ by the sign of $\int$

Solved Examples

1. $\lim \limits _{\mathrm{n} \rightarrow \infty} \dfrac{1}{\mathrm{n}+1}+\dfrac{1}{\mathrm{n}+2}+\dfrac{1}{\mathrm{n}+3}+—-+\dfrac{1}{2 \mathrm{n}}=$

(a) 1

(b) 2

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

(d) $\log _{e} 4$

2. The value of $\sum \limits _{\mathrm{r}=0}^{\mathrm{n}-1} \dfrac{1}{\sqrt{4 \mathrm{n}^{2}-\mathrm{r}^{2}}}$ as $\mathrm{n} \rightarrow \infty$ is

(a) $\dfrac{\pi}{3}$

(b) $\dfrac{\pi}{6}$

(c) $\dfrac{\pi}{4}$

(d) none of these

3. The value of $\lim \limits _{\mathrm{n} \rightarrow \infty} \dfrac{\left(1^{2}+2^{2}+3^{2}+-+\mathrm{n}^{2}\right)\left(1^{3}+2^{3}+-+\mathrm{n}^{3}\right)}{1^{6}+2^{6}+—–+\mathrm{n}^{6}}$

(a) 1

(b) $\dfrac{7}{12}$

(c) $\dfrac{5}{6}$

(d) none

4. $\int _{0}^{1}\left[\dfrac{2}{\mathrm{e}^{\mathrm{x}}}\right] \mathrm{dx}$ is equal to

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

(b) $\mathrm{e}^{2}$

(c) 0

(d) $\dfrac{2}{\mathrm{e}}$

5. The value of $\int _{0}^{\pi / 3}[\sqrt{3} \tan x] d x$ is

(a) $\dfrac{5 \pi}{6}$

(b) $\dfrac{5 \pi}{6}-\tan ^{-1} \dfrac{2}{\sqrt{3}}$

(c) $\dfrac{\pi}{2}-\tan ^{-1} \dfrac{2}{\sqrt{3}}$

(d) none

6. $\int _{-1}^{1}[\mathrm{x}[1+\sin \pi \mathrm{x}]+1] \mathrm{dx}$ is equal to

(a) 2

(b) 0

(c) 1

(d) none

7. $\int _{0}^{2}\left[\mathrm{x}^{2}-1\right] \mathrm{dx}$ is equal to

(a) $3-\sqrt{3}-\sqrt{2}$

(b) 2

(c) 1

(d) $3-2 \sqrt{3}-3 \sqrt{2}$

Exercise

1. $\lim \limits _{\mathrm{n} \rightarrow \infty} \dfrac{1^{\mathrm{P}}+2^{\mathrm{P}}+3^{\mathrm{P}}+……+\mathrm{n}^{\mathrm{P}}}{\mathrm{n}^{\mathrm{P}}+1}=$

(a) $\dfrac{1}{\mathrm{P}+1}$

(b) $\dfrac{1}{1-\mathrm{P}}$

(c) $\dfrac{1}{\mathrm{P}}-\dfrac{1}{\mathrm{P}-1}$

(d) $\dfrac{\mathrm{P}}{\mathrm{P}+1}$

2. Let a,b,c be non-zero real numbers such that $\int _{0}^{1}\left(1+\cos ^{8} \mathrm{x}\right)\left(a x^{2}+\mathrm{bx}+\mathrm{c}\right) \mathrm{dx}$

$=\int _{0}^{2}\left(1+\cos ^{8} \mathrm{x}\right)\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right) \mathrm{dx}$. Then the quadratic equation $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0$ has

(a) no root in $(0,2)$

(b) at least one root in $(1,2)$

(c) a double root in $(0,2)$

(d) two imaginary roots

3. For every function $f(\mathrm{x})$ which is twice differentiable these will be good approximation of

$\begin{aligned} & \int _{\mathrm{a}}^{\mathrm{b}} f(\mathrm{x}) \mathrm{dx}=\dfrac{\mathrm{b}-\mathrm{a}}{2}(f(\mathrm{a})+f(\mathrm{~b})), \text { for more accurate results for } \mathrm{c} \in(\mathrm{a}, \mathrm{b}) \\ & \mathrm{F}(\mathrm{c})=\dfrac{\mathrm{c}-\mathrm{a}}{2}(f(\mathrm{a})-f(\mathrm{c}))+\dfrac{\mathrm{b}-\mathrm{c}}{2}(f(\mathrm{~b})-f(\mathrm{c})) \quad \text { where } \quad \mathrm{c}=\dfrac{\mathrm{a}+\mathrm{b}}{2} \\ & \int _{\mathrm{a}}^{\mathrm{b}} f(\mathrm{x}) \mathrm{dx}=\dfrac{\mathrm{b}-\mathrm{a}}{4}(f(\mathrm{a})+f(\mathrm{~b})+2 f(\mathrm{c})) \mathrm{dx} \end{aligned}$

i. Good approximation of $\int _{0}^{\pi / 2} \sin x d x$ is

(a) $\dfrac{\pi}{4}$

(b) $\dfrac{\pi(\sqrt{2}+1)}{4}$

(c) $\dfrac{\pi(\sqrt{2}+1)}{8}$

(d) $\dfrac{\pi}{8}$

ii. If $f^{\prime \prime}(\mathrm{x})<0, \forall \mathrm{x} \in(\mathrm{a}, \mathrm{b})$ & $ (\mathrm{c}, f(\mathrm{c}))$ is point of maximum where $\mathrm{c} \in(\mathrm{a} . \mathrm{b})$ then $f^{\prime}(\mathrm{c})$ is

(a) $\dfrac{f(\mathrm{~b})-f(\mathrm{a})}{\mathrm{b}-\mathrm{a}}$

(b) $3\left(\dfrac{f(\mathrm{~b})-f(\mathrm{a})}{\mathrm{b}-\mathrm{a}}\right)$

(c) $2\left(\dfrac{f(\mathrm{~b})-f(\mathrm{a})}{\mathrm{b}-\mathrm{a}}\right)$

(d) 0

iii. If $\lim \limits _{\mathrm{t} \rightarrow \mathrm{a}} \dfrac{\int \limits _{\mathrm{a}}^{\mathrm{t}} f(\mathrm{x}) \mathrm{dx}-\dfrac{\mathrm{t}-\mathrm{a}}{2}(f(\mathrm{t})+f(\mathrm{a}))}{(\mathrm{t}-\mathrm{a})^{3}}=0$, then degree of polynomial function $f(\mathrm{x})$ at most is

(a) 0

(b) 1

(c) 3

(d) 2

4. Match the following:-

5. $\int _{1}^{3} \dfrac{d x}{x^{2}+[x]^{2}+1-2 x[x]}=$

(a) $\pi$

(b) $3 \pi$

(c) $8 \pi$

(d) $\dfrac{\pi}{2}$

6.* If $I _{n}=\int _{-\pi}^{\pi} \dfrac{\sin n x}{\left(1+\pi^{x}\right) \sin x} d x, n=0,1,2, \ldots \ldots$ then

(a) $I _{n}=I _{n+2}$

(b) $\sum \limits _{\mathrm{m}=1}^{10} \mathrm{I} _{2 \mathrm{~m}+1}=10 \pi$

(c) $\sum \limits _{\mathrm{m}=1}^{10} \mathrm{I} _{2 \mathrm{~m}}=0$

(d) $I _{n}=I _{n+1}$

7. All the values of a for which $\int _{1}^{2}\left(a^{2}+(4-4 a) x+4 x^{3}\right) d x \leq 12$ are given by

(a) $\mathrm{a} \leq 4$

(b) $a=3$

(c) $0 \leq \mathrm{a}<3$

(d) none of these

8. Value of $\int _{0}^{2}\left[x^{2}-x+1\right] d x$ is

(a) $\dfrac{7+\sqrt{5}}{2}$

(b) $\dfrac{5-\sqrt{5}}{2}$

(c) $\dfrac{7-\sqrt{5}}{2}$

(d) none of these

9. If $I=\int _{0}^{1} \dfrac{d x}{\left(5+2 x-2 x^{2}\right)\left(1+e^{2-4 x}\right)}$ and $I _{1}=\int _{0}^{1} \dfrac{d x}{5+2 x-2 x^{2}} d x$ then

(a) $\mathrm{I}=\dfrac{1}{2} \mathrm{I} _{1}$

(b) $\mathrm{I}=2 \mathrm{I} _{1}$

(c) $3 \mathrm{I}=5 \mathrm{I} _{1}$

(d) none of these

10. Match the following:-

11. $\int _{0}^{2}\left[\tan ^{-1} x\right] d x+\int _{0}^{2}\left[\cot ^{-1} x\right] d x=$

(a) $1-\cot 2$

(b) $1+\cot 2$

(c) $2(1+\cot 2)$

(d) $2(1-\cot 2)$

12.* Match the following

([.] denotes the greatest integer function)