Mathematics

# The value of the definite integral $\int _{ 0 }^{ \pi /2 }{ \sin { x } \sin { 2x } \sin { 3x } dx }$ is equal to:

$\cfrac{1}{6}$

##### SOLUTION
$\int_0^{\pi/2}\sin x\sin 2x\sin 3x dx$

$\Rightarrow \dfrac{1}{2}\int_0^{\pi/2}2\sin x\sin 2x\sin 3x dx$

$\Rightarrow \dfrac{1}{2}\int_0^{\pi/2}2\sin x\sin 3x\sin 2x dx$

We know that       $2\sin A \sin B=\cos(A-B)-\cos (A+B)$

$\Rightarrow \dfrac{1}{2}\int_0^{\pi/2}(\cos (x-3x)-\cos(x+3x))\sin 2x dx$

$\Rightarrow \dfrac{1}{2}\int_0^{\pi/2}(\cos 2x-\cos 4x)\sin 2x dx$

$\Rightarrow \dfrac{1}{2}\int_0^{\pi/2} \sin 2x \cos 2x dx-\dfrac{1}{2}\int _0^{\pi /2}\sin 2x \cos 4x dx$

$\Rightarrow \dfrac{1}{4}\int_0^{\pi/2} 2\sin 2x \cos 2x dx-\dfrac{1}{4}\int _0^{\pi /2}2\sin 2x \cos 4x dx$

We know that   $2\sin A \cos B=\sin(A+B)+\sin (A-B)$

$\Rightarrow \dfrac{1}{4}\int_0^{\pi/2} (\sin (2x+2x)+\sin (2x-2x))dx-\dfrac{1}{4}\int _0^{\pi /2}(\sin (2x+4x)+\sin (2x-4x)) dx$

$\Rightarrow \dfrac{1}{4}\int_0^{\pi/2} \sin 4xdx-\dfrac{1}{4}\int _0^{\pi /2}(\sin 6x-\sin 2x) dx$

$\Rightarrow \dfrac{1}{4}\int_0^{\pi/2} \sin 4xdx-\dfrac{1}{4}\int _0^{\pi /2}\sin 6x dx+\dfrac{1}{4}\int_{0}^{\pi/4}\sin 2x dx$

$\Rightarrow \dfrac{1}{4}[\dfrac{\sin 4x}{4}]_0^{\pi/4}-\dfrac{1}{4}[\dfrac{-\sin 6x}{6}]_0^{\pi/4}+\dfrac{1}{4}[\dfrac{-\cos 2x}{2}]_0^{\pi/4}$

$\Rightarrow \dfrac{-1}{12}+\dfrac{1}{4}=\dfrac{1}{6}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Hard
Integrate  $\int_{0}^{2[x]} \left \{\dfrac {x}{2}\right \} dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int { \cfrac { dx }{ (4{ x }^{ 2 }-4x+3) } } =$?
• A. $\cfrac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \cfrac { 2x-1 }{ \sqrt { 2 } } \right) } +C$
• B. $-\cfrac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \cfrac { 2x-1 }{ \sqrt { 2 } } \right) } +C$
• C. none of these
• D. $\cfrac { 1 }{ 2\sqrt { 2 } } \tan ^{ -1 }{ \left( \cfrac { 2x-1 }{ \sqrt { 2 } } \right) } +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Solve $\displaystyle \int_0^{\pi/2} \dfrac{x\,\,\sin\,x \,cos\,x}{cos^4x+\sin^4x}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\int\limits_0^\infty {\dfrac{{{x^2} + 1}}{{{x^4} + 10{x^2} + 9}}\;{\text{dx}}\;{\text{ = }}}$
• A. $\pi$
• B. $\dfrac{\pi }{2}$
• C. $\dfrac{\pi }{3}$
• D. $\dfrac{\pi }{6}$

Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$