Mathematics

# Evaluate : $\displaystyle\int {\dfrac{{{x^5}}}{{{x^2} + 9}}}dx$

##### SOLUTION

Consider the given integral.

$I=\displaystyle\int{\dfrac{{{x}^{5}}}{{{x}^{2}}+4}}dx$

Let $t={{x}^{2}}+4$

$\dfrac{dt}{dx}=2x+0$

$\dfrac{dt}{2}=xdx$

Therefore,

$I=\dfrac{1}{2}\displaystyle\int{\dfrac{{{\left( t-4 \right)}^{2}}}{t}}dt$

$I=\dfrac{1}{2}\displaystyle\int{\dfrac{{{t}^{2}}+16-8t}{t}}dt$

$I=\dfrac{1}{2}\displaystyle\int{\left( t+\dfrac{16}{t}-8 \right)}dt$

$I=\dfrac{1}{2}\left[ \dfrac{{{t}^{2}}}{2}+16\log \left( t \right)-8t \right]+C$

On putting the value of $t$, we get

$I=\dfrac{1}{2}\left[ \dfrac{{{\left( {{x}^{2}}+4 \right)}^{2}}}{2}+16\log \left( {{x}^{2}}+4 \right)-8\left( {{x}^{2}}+4 \right) \right]+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Single Correct Hard
Evaluate : $\displaystyle\int _{ \tfrac{ \pi }{ 6 } }^{ \tfrac{ \pi }{ 3 } }{ \dfrac { dx }{ 1+\sqrt { \tan { x } } } }$
• A. $\dfrac{\pi}{4}$
• B. $\dfrac{\pi}{8}$
• C. $\dfrac{\pi}{6}$
• D. $\dfrac{\pi}{12}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve $\displaystyle \int _{ 0 }^{ \infty }{ \frac { x\log { x }}{ { \left( 1+{ x }^{ 2 } \right) }^{ 2 } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate $\displaystyle\int^{\pi/3}_{\pi/6}\dfrac{dx}{1+\sqrt{\tan x}}$.
• A. $\cfrac{7\pi}{12}$
• B. $\cfrac{5\pi}{12}$
• C. None of these
• D. $\cfrac{\pi}{12}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The value of $\int {\left( {x - 1} \right){e^{ - x}}}$ is equal to
• A. $- x{e^x} + C$
• B. $x{e^x} + C$
• C. $x{e^{ - x}} + C$
• D. $- x{e^{ - x}} + C$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$