Mathematics

# Let $\displaystyle f\left ( x \right )=\frac{\sin x}{x},then\:\int_{0}^{\pi /2}f\left ( x \right )f\left ( \frac{\pi }{2}-x \right )dx=$

$\displaystyle \frac{2}{\pi }\int_{0}^{\pi }f\left ( x \right )dx$

##### SOLUTION
$\displaystyle l=\int_{0}^{n/2}\frac{\sin x}{x}\frac{\cos x}{\left ( \dfrac{\pi }{2}-x \right )}dx$

$\displaystyle \pi =\int_{0}^{\pi/2 }\sin 2x\left [ \frac{1}{x} +\frac{1}{\pi /2-x}\right ]dx$

$\displaystyle =\int_{0}^{\pi /2}\frac{\sin 2x}{x}dx+\int_{0}^{\pi /2}\frac{\sin 2x}{\pi /2-x}dx$

$\displaystyle =\int_{0}^{\pi /2}\frac{\sin 2x}{x}dx+\int_{0}^{\pi /2}\frac{\sin 2x}{x}dx=4\int_{0}^{\pi /2}\frac{\sin 2x}{2x}dx$

$\displaystyle \frac{\pi }{2}=\int_{0}^{\pi }\frac{\sin t}{t}dt$ [Substitute $2x=t$]

$\displaystyle l=\frac{2}{\pi }\int_{0}^{\pi }\frac{\sin x}{x}dx$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int_{-1}^{1}\frac{1}{(1+x^{2})^{2}}dx=?$
• A. $\displaystyle \frac{\pi}{4}-\frac{1}{2}$
• B. $\displaystyle \frac{\pi}{8}$
• C. $\displaystyle \frac{\pi}{16}$
• D. $\displaystyle \frac{\pi}{4}+\frac{1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Find $\displaystyle \int \frac{dx}{x\left ( x^{3}+1 \right )^{2}}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle \int \dfrac 32 x^{1/2}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Multiple Correct Hard
The value of $\displaystyle \int_{-\sqrt{2}}^{\sqrt{2}}\frac{2x^{7}+3x^{6}-10x^{5}-7x^{3}-12x^{2}+x+1}{x^{2}+2}\:dx$ is

• A. $0$
• B. $\displaystyle \frac{-14}{5}\sqrt{2}+\frac{\pi }{2\sqrt{2}}$
• C. $\displaystyle \frac{-16}{5}\sqrt{2}+\frac{\pi }{2\sqrt{2}}$
• D. $2 \displaystyle \int_{0}^{\sqrt{2}}\frac{3x^{6}-12x^{2}+1}{x^{2}+2}\:dx$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$