Mathematics

# Solve:$\int\limits_0^{\pi /6} {\dfrac{{\cos 2x}}{{{{\left( {\cos x - \sin x} \right)}^2}}}dx}$

$- \log \left( {\dfrac{{\sqrt 3 - 1}}{2}} \right)$

##### SOLUTION
$\int _{ 0 }^{ \pi /6 }{ \dfrac { \cos 2x }{ \left( \cos x-\sin x \right) ^{ 2 } } dx } \\ \int _{ 0 }^{ \pi /6 }{ \dfrac { \cos ^{ 2 } x-\sin ^{ 2 } x }{ \left( \cos x-\sin x \right) ^{ 2 } } dx } \\ \int _{ 0 }^{ \pi /6 }{ \dfrac { (\cos x-\sin x)(\cos x+\sin x) }{ \left( \cos x-\sin x \right) ^{ 2 } } dx } \\ \int _{ 0 }^{ \pi/6 }{ \dfrac { (\cos x+\sin x) }{ (\cos x-\sin x) } dx } \\ put\quad t=\cos x-\sin x\\ Hence, dt=-(\cos x+\sin x)dx\\ -\displaystyle\int _{ 1 }^{ \frac { \sqrt { 3 } -1 }{ 2 }}{\dfrac {dt }{t}} \\ -\bigg[\log { t } \bigg]_{ 1 }^{ \dfrac { \sqrt { 3 } -1 }{ 2 } }\\ -\left[ \log { \left( \dfrac { \sqrt { 3 } -1 }{ 2 } \right) -\log { \left( 1 \right) } } \right] \\ - \log { \left( \dfrac { \sqrt { 3 } -1 }{ 2 } \right) } \quad \quad \quad \quad \because \log { \left( 1 \right) =0 } \\ \\$

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

#### Realted Questions

Q1 Subjective Medium
The value of $\displaystyle \int _{0}^{1}x^2+2 dx$ is equal to

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int x^{3}\tan ^{-1}x\:dx.$
• A. $\displaystyle \frac{1}{2}\left [ \left ( x^{4}-1 \right )\tan ^{-1}x-\frac{x^{2}}{3}+x \right ].$
• B. $\displaystyle \frac{1}{2}\left [ \left ( x^{4}+1 \right )\tan ^{-1}x-\frac{x^{2}}{3}+x \right ].$
• C. $\displaystyle \frac{1}{4}\left [ \left ( x^{4}-1 \right )\tan ^{-1}x-\frac{x^{1}}{3}+x \right ].$
• D. $\displaystyle \frac{1}{4}\left [ \left ( x^{4}-1 \right )\tan ^{-1}x-\frac{x^{2}}{3}+x \right ].$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate: $\int { \dfrac { { x }^{ 3 }\sin ^{ -1 }{ { x }^{ 2 } } }{ \sqrt { 1-{ x }^{ 4 } } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Solve :
$\displaystyle \int { \left( x+2 \right) \sqrt { 3x+5 } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
Repeated application of integration by parts gives us the reduction formula, if the integrand is dependent on a natural number $n$.

If $\displaystyle \int \frac {\cos^m x}{\sin^n x} dx = \frac {\cos^{m - 1}x}{(m - n) \sin^{n - 1} x} + A \int \frac {\cos^{m - 2} x}{\sin^n x} dx + C$, then $A$ is equal to
• A. $\displaystyle \frac {m}{m + n}$
• B. $\displaystyle \frac {m - 1}{m + n}$
• C. $\displaystyle \frac {m}{m + n - 1}$
• D. $\displaystyle \frac {m - 1}{m - n}$