Mathematics

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


SOLUTION

Consider the given integral.

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

$$I=\int{\dfrac{3{{x}^{2}}}{{{\left( {{x}^{3}} \right)}^{2}}+1}}dx$$

Let $$t={{x}^{3}}$$

$$dt=3{{x}^{2}}dx$$

Therefore,

$$ I=\int{\dfrac{dt}{{{t}^{2}}+1}} $$

$$ I={{\tan }^{-1}}t+C $$

On putting the value of $$t$$, we get

$$I={{\tan }^{-1}}\left( {{x}^{3}} \right)+C$$

Hence, this is the answer.

View Full Answer

Its FREE, you're just one step away


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

Realted Questions

Q1 Single Correct Medium
The value of $$\displaystyle  \int_0^1 \sqrt{x} \cdot e^{\sqrt{x}} dx $$ is equal to 
  • A. $$ \dfrac {(e-2)}{2} $$
  • B. $$ 2e-1$$
  • C. $$ 2 (e-1) $$
  • D. $$ \dfrac {e-1}{2} $$
  • E. $$ 2 ( e-2) $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
 $$\displaystyle \int e^{2x-3}+7^{4-3(x/2)}+\sin \left ( 3x-\frac{1}{2} \right )+\cos \left ( \frac{2}{5}x-2 \right )+a^{3x+2}dx$$  is 

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
Evaluate $$\displaystyle\int^2_0\dfrac{dx}{\sqrt{4-x^2}}$$
  • A. $$1$$
  • B. $$\sin^{-1}\dfrac{1}{2}$$
  • C. $$\dfrac{\pi}{4}$$
  • D. None of these

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
$$\displaystyle\lim_{n\rightarrow \infty}\dfrac{3}{n}\left\{1+\sqrt{\dfrac{n}{n+3}}+\sqrt{\dfrac{n}{n+6}}+\sqrt{\dfrac{n}{n+9}}+...….+\sqrt{\dfrac{n}{n+3(n-1)}}\right\}=?$$
  • A. Does not exist
  • B. $$1$$
  • C. $$3$$
  • D. $$2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer