Mathematics

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


ANSWER

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


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 $$u=x^3$$

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

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

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

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

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

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

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.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\sum _ { i = 1 } ^ { 6 } x ^ { i } e ^ { x } + c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
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 $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer