Mathematics

Solve $$\displaystyle\int {{{\left( {{x^2} + \dfrac{1}{{{x^2}}}} \right)}^6}dx} $$


SOLUTION

Consider the given integral.


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


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


$$ I=\displaystyle\int{\left( {{x}^{4}}+\dfrac{1}{{{x}^{4}}}+2 \right)\left( {{x}^{4}}+\dfrac{1}{{{x}^{4}}}+2 \right)\left( {{x}^{4}}+\dfrac{1}{{{x}^{4}}}+2 \right)}dx $$


$$ I=\displaystyle\int{\left( {{x}^{4}}+\dfrac{1}{{{x}^{4}}}+2 \right)\left( {{x}^{8}}+\dfrac{1}{{{x}^{8}}}+4{{x}^{4}}+\dfrac{4}{{{x}^{4}}}+8 \right)}dx $$


$$ I=\displaystyle\int{\left( {{x}^{12}}+\dfrac{15}{{{x}^{4}}}+\dfrac{6}{{{x}^{8}}}+\dfrac{1}{{{x}^{12}}}+6{{x}^{8}}+15{{x}^{4}}+20 \right)}dx $$


$$ I=\left( \dfrac{{{x}^{13}}}{13}-\dfrac{6}{{{x}^{3}}}-\dfrac{6}{7{{x}^{7}}}-\dfrac{1}{11{{x}^{11}}}+\dfrac{2{{x}^{9}}}{3}+3{{x}^{5}}+20x \right)+C $$


 


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
$$\int_0^{\pi/2} \dfrac{tan^7 x}{cot^7 x + tan^7 x} dx$$ is equal to
  • A. $$\dfrac{\pi}{6}$$
  • B. $$\dfrac{\pi}{2}$$
  • C. $$\dfrac{\pi}{3}$$
  • D. $$\dfrac{\pi}{4}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$\int_{-1}^{1 / 2} \dfrac{e^{x}\left(2-x^{2}\right) d x}{(1-x) \sqrt{1-x^{2}}}$$ is equal to
  • A. $$\dfrac{\sqrt{e}}{2}(\sqrt{3}+1)$$
  • B. $$\dfrac{\sqrt{3 e}}{2}$$
  • C. $$\sqrt{\dfrac{e}{3}}$$
  • D. $$\sqrt{3 e}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Hard
Evaluate $$\displaystyle \int_{0}^{\pi}\mathrm{e}^{|\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{x}|}(2\sin(\frac{1}{2}\cos \mathrm{x})+3\cos(\frac{1}{2}\cos \mathrm{x})) \sin x dx $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the following integral
$$\int { \cfrac { \sin { \left( x-\alpha \right)  }  }{ \sin { \left( x+\alpha \right)  }  }  } dx$$

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