Mathematics

Evaluate :
$$\displaystyle \int x^{3}\sqrt{1-x^{8}}dx$$


SOLUTION
$$I=\int { { x }^{ 3 }\sqrt { 1-{ x }^{ 8 } } dx } =\int { { x }^{ 3 }\sqrt { 1-{ ({ x }^{ 4 }) }^{ 2 } } dx } $$
Putting $${ x }^{ 4 }=t$$
Differentiating both sides
$$4{ x }^{ 3 }dx=dt\\ dx=\cfrac { dt }{ 4{ x }^{ 3 } } $$
$$I=\cfrac { 1 }{ 4 } \int { \sqrt { 1-{ t }^{ 2 } } dt } =\cfrac { 1 }{ 4 } \{ [\cfrac { t }{ 2 } \sqrt { 1-{ t }^{ 2 } } +\cfrac { 1 }{ 2 } \sin ^{ -1 }{ t } ]+C\} \\ \because \int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } dx } =\cfrac { x }{ 2 } \sqrt { { a }^{ 2 }-{ x }^{ 2 } } +\cfrac { { a }^{ 2 } }{ 2 } \sin ^{ -1 }{ \cfrac { x }{ a }  } +C$$
Putting the value $$t={ x }^{ 4 }$$
$$I=\cfrac { 1 }{ 8 } [{ x }^{ 4 }\sqrt { 1-{ x }^{ 8 } } +\sin ^{ -1 }{ { x }^{ 4 } } ]+C$$
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 86
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
The quadratic polynomial $$p(x)$$ has the following properties: $$p(x)\ge 0$$ for all real number, $$p(1)=$$ and $$p(2)=2$$ value of $$p(0)+p(3)$$ is equal to 
  • A. $$9$$
  • B. $$8$$
  • C. $$None \ of \ these$$
  • D. $$10$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate the integral   $$\displaystyle \int_0^1\sin^{-1}\left (\frac {2x}{1+x^2}\right )dx$$   using substitution.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
Evaluate $$\displaystyle \int_{}^{} {x\sqrt {\frac{{{a^2} - {x^2}}}{{{a^2} + {x^2}}}} dx  } $$
  • A. $$\displaystyle \frac{1}{2}{a^2}{\cos ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \frac{1}{2}\sqrt {{a^4} + {x^4}} + C$$
  • B. $$\displaystyle \frac{1}{2}{\sin ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \sqrt {{a^4} + {x^4}} + C$$
  • C. $$\displaystyle \frac{1}{2}{\cos ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \frac{1}{2}\sqrt {{a^4} - {x^4}} + C$$
  • D. $$\displaystyle \frac{1}{2}{a^2}{\sin ^{ - 1}}\left( {\frac{{{x^2}}}{{{a^2}}}} \right) + \frac{1}{2}\sqrt {{a^4} - {x^4}} + C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
$$\int \dfrac {(x^{2}+2)}{(x^{2}+1)(x^{2}+4)}dx$$ (for $$0 < x < 1)=$$
  • A. $$\dfrac {1}{2}\tan^{-1}\left(-\dfrac {3x}{x^{2}-2}\right)+c$$
  • B. $$\dfrac {1}{2}\tan^{-1}\left(\dfrac {2x}{x^{2}+ 2}\right)+c$$
  • C. $$none\ of\ these$$
  • D. $$\dfrac {1}{2}\tan^{-1}\left(\dfrac {2x}{x^{2}-2}\right)+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Evaluate $$\displaystyle \int_0^{10\pi}|\sin\,x|dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer