Mathematics

Evaluate:
$$\displaystyle \int \dfrac {dx}{x(x^{3}+8)}$$


SOLUTION
We have,
$$I=\int { \dfrac { dx }{ x\left( { x }^{ 3 }+8 \right)  }  } $$
$$I=\int { \dfrac { { x }^{ 2 }dx }{ { x }^{ 3 }\left( { x }^{ 3 }+8 \right)  }  } $$

Let $${ x }^{ 3 }=t\quad \Rightarrow 3{ x }^{ 2 }dx=dt$$
$$\Rightarrow { x }^{ 2 }dx=\dfrac { dt }{ 3 } $$

$$\therefore$$    $$I=\int { \dfrac { 1 }{ 3t\left( t+8 \right)  }  } dt$$
$$=\dfrac { 1 }{ 3 } \int { \dfrac { 1 }{ 8 } \dfrac { \left( t+8-t \right)  }{ \left( t \right) \left( t+8 \right)  }  } dt$$
$$=\dfrac { 1 }{ 24 } \int { \left( \dfrac { 1 }{ t } -\dfrac { 1 }{ t+8 }  \right)  } dt$$
$$=\dfrac { 1 }{ 24 } \log\left( t \right) -log\left( t+8 \right) +c$$
$$=\dfrac { 1 }{ 24 } \log\left| \dfrac { t }{ t+8 }  \right| +c$$
$$=\dfrac { 1 }{ 24 } \log\left| \dfrac { { x }^{ 3 } }{ { x }^{ 3 }+8 }  \right| +c$$

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 Subjective Medium
Evaluate the following integrals:
$$\int { \cfrac { \cos { 2x }  }{ \sqrt { \sin ^{ 2 }{ 2x } +8 }  }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
$$\displaystyle \int \dfrac{\sec^2 x}{(\sec x + \tan  x)^{9/2}} dx$$ equals
  • A. $$- \dfrac{1}{(\sec x + \tan x)^{11/2}} \left \{ \dfrac{1}{11} - \dfrac{1}{7} (\sec x + \tan x)^2 \right \} + K$$
  • B. $$\dfrac{1}{(\sec x + \tan x)^{11/2}} \left \{ \dfrac{1}{11} - \dfrac{1}{7} (\sec x + \tan x)^2 \right \} + K$$
  • C. None of the above
  • D. $$-\dfrac{1}{(\sec x + \tan x)^{11/2}} \left \{ \dfrac{1}{11} + \dfrac{1}{7} (\sec x + \tan x) \right \} + K$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
$$\int \frac{1}{3x^2+ 13x -10}dx$$
  • A. $$\frac{1}{17}[log(\frac{30}{17}+\frac{6x}{17})-log(\frac{-4}{17}+\frac{6x}{17})]+C$$
  • B. $$\frac{-1}{17}[log(\frac{30}{17}+\frac{6x}{17})+log(\frac{-4}{17}+\frac{6x}{17})]+C$$
  • C. $$\frac{-1}{17}[log(\frac{30}{17}+\frac{6x}{17})-log(\frac{-4}{17}-\frac{6x}{17})]+C$$
  • D. $$\frac{-1}{17}[log(\frac{30}{17}+\frac{6x}{17})-log(\frac{-4}{17}+\frac{6x}{17})]+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Multiple Correct Hard
The value of $$\displaystyle \int_{0}^{\infty} \frac {dx}{1 + x^4}$$ is
  • A. same as that of $$\displaystyle \int_{0}^{\infty} \frac {x^2 + 1dx}{1 + x^4}$$
  • B. $$\displaystyle \frac {\pi}{\sqrt{2}}$$
  • C. $$\displaystyle \frac {\pi}{2\sqrt{2}}$$
  • D. same as that of $$\displaystyle \int_{0}^{\infty} \frac {x^2 \: dx}{1 + x^4}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Evaluate:
$$\displaystyle\int \dfrac{1}{e^x+e^{-x}}dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer