Mathematics

# Evaluate:$\int \frac{1}{x^{4}-1}dx$

##### SOLUTION
$\begin{array}{l} \int _{ }^{ }{ \dfrac { 1 }{ { { x^{ 4 } }-1 } } { dx } } =\int _{ }^{ }{ \dfrac { 1 }{ { \left( { x-1 } \right) \left( { x+1 } \right) \left( { { x^{ 2 } }+1 } \right) } } { dx } } \\ =\int _{ }^{ }{ \left( { -\dfrac { 1 }{ { 2\left( { { x^{ 2 } }+1 } \right) } } -\dfrac { 1 }{ { 4\left( { x+1 } \right) } } +\dfrac { 1 }{ { 4\left( { x-1 } \right) } } } \right) } dx \\ =-\dfrac { { \ln { \left( { x+1 } \right) } } }{ 4 } -\dfrac { { { { \tan }^{ -1 } }\left( x \right) } }{ 2 } +\dfrac { { \ln { \left( { x-1 } \right) } } }{ 4 } +C \end{array}$

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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 Medium
Evaluate:$\displaystyle \int \frac{dx}{\sqrt{x-x^{2}}}$
• A. $\dfrac12 \sin^{-1}(2x-1)+C$
• B. $2 \sin^{-1}(2x-1)+C$
• C. none of these
• D. $\sin^{-1}(2x-1)+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following definite integral:
$\displaystyle \int_{e}^{e^2} \left\{\dfrac {1}{\log x} -\dfrac {1}{(\log x)^2}\right\} dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle\int \frac{xdx}{(1+x)^{3/2}}$ is equal to.
• A. $\displaystyle\frac{(2+x)}{\sqrt{1+x}}+C$
• B. $\displaystyle\frac{3}{2}\frac{3}{\sqrt{1+x}}+C$
• C. $\displaystyle\frac{3}{2}\frac{(2+x)}{\sqrt{1+x}}+C$
• D. $2\displaystyle\frac{(2+x)}{\sqrt{1+x}}+C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $\displaystyle \frac{(x-1)^2}{x^3+x}=\frac{A}{x}+\frac{Bx+C}{x^2+1}$, then $A=..., B=..., C=...$
• A. $A=1,B=0,C=2$
• B. $A=0,B=1,C=-2$
• C. $A=0,B=1,C=2$
• D. $A=1,B=0,C=-2$

Let $F: R\rightarrow R$ be a thrice differential function. Suppose that $F(1) = 0, F(3) = -4$ and $F'(x)<0$ for all $x\in\left(\dfrac{1}{2},3\right)$. Let $f(x) = xF(x)$ for all $x\in R$.