Mathematics

# Evaluate :$\int { { sin }^{ 5 }x.dx }$

##### SOLUTION
$\displaystyle\int{{\sin}^{5}{x}dx}$
$=\displaystyle\int{{\sin}^{4}{x}\sin{x}dx}$
$=\displaystyle\int{{\left(1-{\cos}^{2}{x}\right)}^{2}\sin{x}dx}$
Let $u=\cos{x}\Rightarrow du=-\sin{x}dx$
$=-\displaystyle\int{{\left(1-{u}^{2}\right)}^{2}du}$
$=-\displaystyle\int{\left(1+{u}^{4}-2{u}^{2}\right)du}$
$=-\left[u+\dfrac{{u}^{5}}{5}-\dfrac{2{u}^{3}}{3}\right]+c$
$=-\cos{x}-\dfrac{{\cos}^{5}{x}}{5}+\dfrac{2{\cos}^{3}{x}}{3}-c$
$=-\dfrac{{\cos}^{5}{x}}{5}+\dfrac{2{\cos}^{3}{x}}{3}-\cos{x}-c$

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

#### Realted Questions

Q1 Single Correct Hard
A function $f\left( X \right)$ which satisfies the relation $\displaystyle f\left( X \right) ={ e }^{ x }+\int _{ 0 }^{ 1 }{ { e }^{ x }f\left( t \right) } dt$, then $f\left( X \right)$ is
• A. $\left( e-2 \right) e^{ x }$
• B. $2e^{ x }$
• C. $\dfrac { { e }^{ x } }{ 2 }$
• D. $\dfrac { { e }^{ x } }{ 2-e }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle\int \sin x\log \left ( \sec x+\tan x \right )dx.$
• A. $\displaystyle \cos x \log \left ( \sec x+\tan x \right )-x^2+c$
• B. $\displaystyle \sin x \log \left ( \sec x+\tan x \right )-x+c$
• C. $\displaystyle -\sec x \log \left ( \sec x+\tan x \right )-x^2+c$
• D. $\displaystyle -\cos x \log \left ( \sec x+\tan x \right )-x+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle\int_{0}^{\pi}\frac{dx}{1+a^{\cos x}}$ equals

• A. $0$
• B. $\displaystyle \pi$
• C. None of these
• D. $\displaystyle\frac{\pi }{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate: $\int_{1}^{3}\left(2 x^{2}+5 x\right) d x$ as a limit of a sum.

$\int \frac{x}{x^2 + a^2} \;dx$