Mathematics

# $\displaystyle \int_0^{\pi} sin x =\displaystyle \lim_{n \rightarrow \infty} \displaystyle \Sigma_{i=1}^n sin \left(\dfrac{\pi i}{n}\right) \dfrac{\pi}{n}$State whether the above statement is True or False?

True

##### SOLUTION
We have to state whether $\int_{0}^{\pi} sin x dx$ can be represented as $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} sin \left(\dfrac{\pi i}{n}\right)\dfrac{\pi}{n}$ is true or false.

We know that $\int_{a}^{b} f(x) dx=\lim_{n \rightarrow \infty} \left[\dfrac{b-a}{n} \sum_{i=1}^{n} f\left(a+i\dfrac{b-a}{n}\right)\right]$

Here we have $f(x)=sin x,a=0,b=\pi$

Therefore $\int_{0}^{\pi} sin x dx=\lim_{n \rightarrow \infty} \left[\dfrac{\pi-0}{n} \sum_{i=1}^{n} f\left(0+i\dfrac{\pi-0}{n}\right)\right]$

$=\lim_{n \rightarrow \infty} \left[\dfrac{\pi}{n} \sum_{i=1}^{n} f\left(\dfrac{\pi i}{n}\right)\right]$

$=\lim_{n \rightarrow \infty} \left[\dfrac{\pi}{n} \sum_{i=1}^{n} sin \left(\dfrac{\pi i}{n}\right)\right]$

$\int_{0}^{\pi} sin x dx=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} sin \left(\dfrac{\pi i}{n}\right)\dfrac{\pi}{n}$

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TRUE/FALSE Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Single Correct Medium
$\underset{0}{\overset{\pi}{\int}} xf (\sin \, x) dx =$
• A. $\pi \underset{0}{\overset{\frac{\pi}{2}}{\int}} f (\cos \, x ) dx$
• B. $\pi \, \underset{0}{\overset{\pi}{\int}} f(\cos \, x) dx$
• C. $\pi \, \underset{0}{\overset{\pi}{\int}} f (\sin \, x) dx$
• D. $\dfrac{\pi}{2}$$\underset{0}{\overset{\pi}{\int}} f (\sin \, x) dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium

Solve $\int {\dfrac{{{x^5}}}{{\sqrt {1 + {x^2}} }}} \,dx$

• A. $\dfrac{1}{{15}}\sqrt {1 + {x^2}} \left( {3{x^4} + 4{x^2} + 8} \right) + C$
• B. $\sqrt {1 + {x^2}} \left( {3{x^4} + 4{x^2} + 8} \right) + C$
• C. None of these
• D. $\dfrac{1}{{15}}\sqrt {1 + {x^2}} \left( {3{x^4} - 4{x^2} + 8} \right) + C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$I$ : $\displaystyle \int e^{x}(1-\cot x+\cot^{2}x)dx=-e^{x}\cot x+c$

$\displaystyle II:\int e^{x}(\frac{1+x\log x}{x})dx=e^{x}\log x+c$
• A. only I is true
• B. only II is true
• C. neither I nor II are true
• D. both I and II are true

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of the integral $\displaystyle \int _ { 0 } ^ { 1 } \dfrac { d x } { x ^ { 2 } + 2 x \cos \alpha + 1 }$ , where $0 < \alpha < \dfrac { \pi } { 2 } ,$ is equal to
• A. $\sin{\alpha}$
• B. $\alpha \sin {\alpha}$
• C. $\dfrac { \alpha } { 2 } \sin {\alpha}$
• D. $\dfrac { \alpha } { 2 \sin{\alpha} }$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$