Mathematics

# $\displaystyle \overset{\pi/2}{\underset{0}{\int}}{ \dfrac { \sin ^6x }{ \cos ^{ 6 }{ x } +\sin ^{ 6 }{ x } } }dx$ is equal to:

$\dfrac{\pi}{4}$

##### SOLUTION
Let, $I=$$\displaystyle \overset{\pi/2}{\underset{0}{\int}}{ \dfrac { \sin ^6x }{ \cos ^{ 6 }{ x } +\sin ^{ 6 }{ x } } }dx.......(1). or, I=$$\displaystyle \overset{\pi/2}{\underset{0}{\int}}{ \dfrac { \cos ^6x }{ \sin ^{ 6 }{ x } +\cos ^{ 6 }{ x } } }dx$ [ Using property of definite integral].......(2).
Now adding (1) and (2) we get,
$2I=\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}} \ dx$
or, $I=\dfrac{\pi}{4}$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium
Solve $\int_{0}^{\frac{\pi }{2}}{{{\sin }^{2}}x}\cos xdx$
• A. $\dfrac{2}{3}$
• B. $\dfrac{5}{3}$
• C. $-\dfrac{1}{3}$
• D. $\dfrac{1}{3}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Evalaute the integral
$\displaystyle \int_{0}^{1}\sin^{-1}(\frac{2x}{1+x^{2}})dx$
• A. $\displaystyle \frac{\pi}{4}-\log 2$
• B. $\displaystyle \frac{\pi}{2}+\log 2$
• C. $\dfrac{\pi}{4}+\log 2$
• D. $\displaystyle \frac{\pi}{2}-\log 2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int { \sqrt { secx-1 } } dx$ is equal to
• A. $2\ell n(cos\frac { x }{ 2 } +\sqrt { { cos }^{ 2 }\frac { x }{ 2 } -\frac { 1 }{ 2 } ) } +C$
• B. $\ell n(cos\frac { x }{ 2 } +\sqrt { { cos }^{ 2 }\frac { x }{ 2 } -\frac { 1 }{ 2 } ) } +C$
• C. $none$ $of$ $these$
• D. $-2\ell n(cos\frac { x }{ 2 } +\sqrt { { cos }^{ 2 }\frac { x }{ 2 } -\frac { 1 }{ 2 } ) } +C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Find its:-
$\mathop {Lt}\limits_{x \to \infty } \left( {\frac{{1 + {2^4} + {3^4} + ......{n^4}}}{{{n^5}}}} \right) - \mathop {Lt}\limits_{x \to \infty } \left( {\frac{{1 + {2^3} + {3^3} + ......{n^3}}}{{{n^5}}}} \right) =$
• A. $0$
• B. $\frac{1}{5}$
• C. $3$
• D. $\frac{1}{4}$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.