Mathematics

# Evaluate : $\underset{0}{\overset{\frac{\pi}{2}}{\int}} \sin^4x \times \cos^5x. dx$

##### SOLUTION
$\int_{0}^{\frac{\pi}{2}}\sin ^4x\cos ^5xdx$

$=\int_{0}^{\frac{\pi}{2}}\sin ^4x(\cos ^2x)^2\cos xdx$

$=\int_{0}^{\frac{\pi}{2}}\sin ^4x(1-\sin ^2x)^2\cos xdx$

substitute $u=\sin x\Rightarrow du=\cos x dx$

$=\int_{0}^{1}u^4(1-u^2)^2du$

$=\int_{0}^{1}u^4-2u^6+u^8du$

$=\left [ \dfrac{u^5}{5}-2\dfrac{u^7}{7}+\dfrac{u^9}{9} \right ]_0^1$

$=\left [ \dfrac{\sin ^5x}{5}-2\dfrac{\sin ^7x}{7}+\dfrac{\sin ^9x}{9} \right ]_0^1+C$

$=\dfrac{8}{315}-0$

$\int_{0}^{\frac{\pi}{2}}\sin ^4x\cos ^5xdx=\dfrac{8}{315}$

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

#### Realted Questions

Q1 Subjective Medium
$I=\int(2x+3)\sqrt{x-2}\ dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate $\int sec^2x. cosec^2 x dx$ on $I \subset R \ \left( \{ n \pi ; n \in Z \} \cup \left \{ (2n + 1) \dfrac{\pi}{2} : n \in Z \right \} \right )$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Partial fraction from of $\displaystyle \frac{3x+7}{x^2-3x+2}$ is
• A. $\displaystyle \frac{13}{x-2}+\frac{10}{x-1}$
• B. $\displaystyle -\frac{13}{x-2}+\frac{10}{x-1}$
• C. $\displaystyle \frac{11}{x-2}-\frac{10}{x-1}$
• D. $\displaystyle \frac{13}{x-2}-\frac{10}{x-1}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle I = \int \frac{dx}{\left ( x^{2} + a^{2} \right ) \left ( x^{2} + b^{2} \right ) \left ( x^{2} + c^{2} \right )}$, then $\displaystyle I$ equals
• A. $\displaystyle \frac{1}{bc} \tan^{-1} \left ( a \right ) + \frac{1}{ca} \tan^{-1} \left ( b \right ) + \frac{1}{cb} \tan^{-1} \left ( c \right ) + k$
• B. $\displaystyle \frac{1}{b^{2}- c^{2}} \tan^{-1} \left ( a \right ) + \frac{1}{c^{2} - a^{2}} \tan^{-1} \left ( b \right ) + \frac{1}{a^{2} - b^{2}} \tan^{-1} \left ( c \right ) + k$
• C. $\displaystyle \frac{\tan^{-1} a + \tan^{-1} b + \tan^{-1} c} {a^{2} + b^{2} + c^{2}} + k$
• D. none of these

Given that for each $\displaystyle a \in (0, 1), \lim_{h \rightarrow 0^+} \int_h^{1-h} t^{-a} (1 -t)^{a-1}dt$ exists. Let this limit be $g(a)$
In addition, it is given that the function $g(a)$ is differentiable on $(0, 1)$