Mathematics

# Evaluate:$\displaystyle\int_{0}^{\frac{\pi}{2}}{{\sin}^{3}{x}\,dx}$

##### SOLUTION
$\displaystyle\int_{0}^{\frac{\pi}{2}}{{\sin}^{3}{x}\,dx}$
$=\dfrac{1}{4}\displaystyle\int_{0}^{\frac{\pi}{2}}{4{\sin}^{3}{x}\,dx}$
We know that $\sin{3x}=3\sin{x}-4{\sin}^{3}{x}\Rightarrow\,4{\sin}^{3}{x}=3\sin{x}-\sin{3x}$
$=\dfrac{1}{4}\displaystyle\int_{0}^{\frac{\pi}{2}}{\left(3\sin{x}-\sin{3x}\right)dx}$
$=\dfrac{1}{4}\displaystyle\int_{0}^{\frac{\pi}{2}}{3\sin{x}\,dx}-\dfrac{1}{4}\displaystyle\int_{0}^{\frac{\pi}{2}}{\sin{3x}\,dx}$
$=\dfrac{3}{4}\displaystyle\int_{0}^{\frac{\pi}{2}}{\sin{x}\,dx}-\dfrac{1}{4}\displaystyle\int_{0}^{\frac{\pi}{2}}{\sin{3x}\,dx}$
$=\dfrac{3}{4}\left[-\cos{x}\right]_{0}^{\frac{\pi}{2}}-\dfrac{1}{4}\left[\dfrac{-\cos{3x}}{3}\right]_{0}^{\frac{\pi}{2}}$
$=\dfrac{-3}{4}\left[\cos{\dfrac{\pi}{2}}-\cos{0}\right]+\dfrac{1}{12}\left[\cos{\dfrac{3\pi}{2}}-\cos{0}\right]$
$=\dfrac{-3}{4}\left[0-1\right]+\dfrac{1}{12}\left[\cos{\left(\pi+\dfrac{\pi}{2}\right)}-1\right]$
$=\dfrac{3}{4}+\dfrac{1}{12}\left[\cos{\dfrac{\pi}{2}}-1\right]$
$=\dfrac{3}{4}+\dfrac{1}{12}\left[0-1\right]$
$=\dfrac{3}{4}-\dfrac{1}{12}$
$=\dfrac{3\times 3}{4\times 3}-\dfrac{1}{12}$
$=\dfrac{9}{12}-\dfrac{1}{12}$
$=\dfrac{8}{12}=\dfrac{2}{3}$

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

#### Realted Questions

Q1 Single Correct Medium
If $\displaystyle \int { \left( u\cfrac { dv }{ dx } \right) } dx=uv-\int { wdx }$, then $w=$
• A. $\cfrac { du }{ dx } \cfrac { dv }{ dx }$
• B. $\cfrac { d }{ dx } (uv)$
• C. $u\cfrac { dv }{ dx }$
• D. $v\cfrac { du }{ dx }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Let $F\left( x \right) =f\left( x \right) +f\left( \dfrac { 1 }{ x } \right)$, where $\displaystyle f\left( x \right) =\int _{ 1 }^{ x }{ \dfrac { \log { t } }{ 1+t } } dt$. Then $F\left( e \right)$ equals -
• A. $0$
• B. $1$
• C. $2$
• D. $\dfrac { 1 }{ 2 }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle \int _{-a}^{a} x^{3}\sqrt{a^{2}-x^{2}}dx=0$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate $\int x^6 dx$
• A. $7x^7 +C$
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• C. $6x^7 +C$
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1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$