Mathematics

# Evaluate: $\displaystyle\int { \dfrac { { x }^{ 4 }+1 }{ 1+{ x }^{ 6 } } } dx$

${ tan }^{ -1 }(x)+\frac { 1 }{ 3 } { tan }^{ -1 }({ x }^{ 3 })+c$

##### SOLUTION
Given,

$\int \dfrac{x^4+1}{x^6+1}dx$

$=\int \dfrac{x^4+1}{x^6+1}\times \dfrac{x^2+1}{x^2+1}dx$

$=\int \dfrac{(x^6+1)+x^2(x^2+1)}{(x^6+1)(x^2+1)}dx$

$=\int \dfrac{dx}{x^2+1}+\dfrac{1}{3}\int \dfrac{3x^2}{x^6+1}dx$

$=\tan ^{-1}x+\dfrac{1}{3}3\cdot \int \dfrac{x^2}{x^6+1}dx$

substitute $u=x^3$

$=\tan ^{-1}x+\dfrac{1}{3}3\cdot \int \dfrac{1}{3\left(u^2+1\right)}du$

$=\tan ^{-1}x+\dfrac{1}{3}3\cdot \dfrac{1}{3}\cdot \int \dfrac{1}{u^2+1}du$

$=\tan ^{-1}x+\dfrac{1}{3}3\cdot \dfrac{1}{3}\cdot \tan ^{-1}u$

$=\tan ^{-1}x+\dfrac{1}{3}3\cdot \dfrac{1}{3}\cdot \tan ^{-1}x^3$

$=\tan ^{-1}x+\dfrac{1}{3} \tan ^{-1}x^3+C$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 One Word Hard
$\displaystyle \int \frac{x^{2}-3}{x^{3}-2x^{2}-x+2}dx=\frac{A}{1494}\log\left ( C\left | \frac{\left ( x-2 \right )\left ( x-1 \right )^{3}}{x+1} \right | \right )$ then A is equal to.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\displaystyle \int \dfrac{x-3}{(x-1)^{3}} dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int \left( x ^ { 6 } + 7 x ^ { 5 } + 6 x ^ { 4 } + 5 x ^ { 3 } + 4 x ^ { 2 } + 3 x + 1 \right) e ^ { x } d x$ equals
• A. $\sum _ { j = 0 } ^ { 6 } x ^ { j } e ^ { x } + c$
• B. $\sum _ { j = 1 } ^ { 7 } x ^ { j } e ^ { x } + c$
• C. $\sum _ { i = 0 } ^ { 5 } x ^ { i } e ^ { x } + c$
• D. $\sum _ { i = 1 } ^ { 6 } x ^ { i } e ^ { x } + c$

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Hard
If $I= \displaystyle \int_{0}^{\pi }\displaystyle \frac{x^{2}\sin ^{2}x\cos ^{4}x}{x^{2}-3\pi x+3x^{2}} dx$ then the value of $\displaystyle \frac{32}{\pi ^{2}} I+ 298$ is equal

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 $\displaystyle f(x)>0 \forall x \in R$