Mathematics

# If $\int _{ 0 }^{ a }{ f\left( x \right) dx=\int _{ 0 }^{ a }{ f\left( a-x \right) dx } }$, then the value of $\int _{ 0 }^{ \pi }{ xf\left( \sin { x } \right) } dx=$

$\dfrac { \pi }{ 2 } \int _{ 0 }^{ \pi }{ f\left( \sin { x } \right) dx }$

##### SOLUTION
Let $S=\int_{0}^{π} xf(sinx)dx$
$\int_{0}^{π} xf(sinx) dx=\int_{0}^{π} (π-x)f(sin(π-x))dx$
$S=\int_{0}^{π}(π-x)f(sinx)dx$
$S=\int_{0}^{π}πf(sinx)dx-\int_{0}^{π}xf(sinx)dx$
$S=π\int_{0}^{π}f(sinx)dx-S$
$2S=π\int_{0}^{π}f(sinx)dx$
$S=\dfrac{π}{2}\int_{0}^{π}f(sinx)dx$
$\int_{0}^{π} xf(sinx) dx=\dfrac{π}{2}\int_{0}^{π}f(sinx)dx$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \lim_{n \rightarrow \infty} \displaystyle \frac{1}{n}[e^{1/n}+e^{2/n}+\ldots..+e]=?$
• A. $\mathrm{e}+1$
• B. $e$
• C. $2e$
• D. $e -1$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int\frac{\sqrt{1-x^{2}}+\sqrt{1+x^{2}}}{\sqrt{1-x^{4}}}dx=$
• A. $\cosh^{-1}x +\sin^{-1}x +c$
• B. $\cosh^{-1}x +\cos^{-1}x +c$
• C. $\sinh^{-1}x +\cos^{-1}x +c$
• D. $\sinh^{-1}x +\sin^{-1}x +c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle\int \dfrac{\cos x \, dx}{\sin^3x(1+\sin^6x)^{2/3}} = f(x)(1 + \sin^6x)^{1/\alpha} + c$
Where $c$ is a constant of integration, then $\lambda f\left(\dfrac{\pi}{3}\right)$ is equal to:
• A. $\dfrac{9}{8}$
• B. $2$
• C. $-\dfrac{9}{8}$
• D. $-2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Passage Hard
Let $g(x)$ be a function defined on $[0, 7]$ and $g(x)=\int_0^x f(t) dt$, where $y=f(x)$ is the function whose graph is as shown in figure given below, then

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
the value of $\underset { n\rightarrow \infty }{ lim } \sum _{ r=1 }^{ n }{ \frac { 1 }{ \sqrt { { n }^{ 2 }-{ r }^{ 2 }{ } } } }$
• A. ${ { x } }^{ 2 }{ sin }^{ -1 }x$
• B. ${ { x } }^{ }{ sin }^{ -1 }x$
• C. $\frac { { sin }^{ -1 }x }{ x }$
• D. ${ sin }^{ -1 }x$