Mathematics

# Evaluate:$\displaystyle\int\dfrac{1}{x^{1-n}(1+x^{2n})}dx$

##### SOLUTION
Now,
$\displaystyle\int\dfrac{1}{x^{1-n}(1+x^{2n})}dx$
$=\displaystyle\int\dfrac{x^{n-1}}{(1+x^{2n})}dx$
$=\dfrac{1}{n}\displaystyle\int\dfrac{d(x^{n})}{(1+(x^{n})^2)}dx$
$=\dfrac{1}{n}\tan^{-1}x^n+c$ [ Where $c$ is integrating constant]

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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 Hard
If $\displaystyle f(x)=\lim_{n\rightarrow \infty }(2x+4x^{3}+......+2^{n}x^{2n-1})\left ( 0<x<\frac{1}{\sqrt{2}} \right )$, then the value of $\displaystyle\int f(x) dx$ is equal to
$\textbf{Note}$: $c$ is the constant of integration.
• A. $\displaystyle \log\left ( \frac{1}{\sqrt{1-x^{2}}} \right )+c$
• B. $\displaystyle \log\sqrt{1-2x^{2}+x} + c$
• C. None of these
• D. $\displaystyle \log\left ( \frac{1}{\sqrt{1-2x^{2}}} \right )+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of $\displaystyle \int_{\pi ^{3}/27}^{\pi ^{3}/8}\sin x\: dt$, where $\displaystyle t=x^{3},$ is
• A. $\displaystyle \cos \frac{\pi ^{3}}{27}-\cos \frac{\pi ^{3}}{8}$
• B. $\displaystyle \frac{\pi ^{2}}{6}$
• C. none of these
• D. $\displaystyle \frac{\pi ^{2}}{6}+\left ( 3-\sqrt{3} \right )\pi -3$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Find the derivative of $\dfrac{e^{x}}{\sin x}$.
• A. $e^x\text{cosec }x[\cot x+1]$
• B. $e^x\sec x[\cot x+1]$
• C. $e^x\sec x[\cot x-1]$
• D. $e^x\text{cosec }x[1-\cot x]$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\int { \cfrac { { x }^{ 4 } }{ \left( x+2 \right) \left( { x }^{ 2 }+1 \right) } dx }$
How to change this improper function to Rational function.

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium

$\displaystyle \int_{0}^{2\pi}e^{ax}cosbx.dx; \mathrm{a},\ \mathrm{b} \in \mathrm{z}=$
• A. $\displaystyle \frac{b}{a^{2}+b^{2}}(e^{2a\pi}-1)$
• B. $\displaystyle \frac{a}{a^{2}+b^{2}}$
• C. $\displaystyle \frac{b}{a^{2}+b^{2}}$
• D. $\displaystyle \frac{a}{a^{2}+b^{2}}(e^{2a\pi}-1)$