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Integrate the following w.r.t x: 1)1/(x^2(1+x^4)^(3/4)) 2)x^5(a^3+x^3)^(1/2)

Integrate the following w.r.t x:
1)1/(x^2(1+x^4)^(3/4))
2)x^5(a^3+x^3)^(1/2)
 

Grade:11

1 Answers

Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
9 years ago
Dear student, please find the solutions:
  1. \int \frac{1}{x^{2}\left ( 1+x^{4} \right )^{\frac{3}{4}}}dx=\int \frac{sec^{2}\Theta }{2x^{3}sec^{\frac{3}{2}}\Theta }d\Theta =\frac{1}{2}\int \frac{cos\Theta }{sin^{\frac{3}{2}}\Theta }d\Theta =-\frac{1}{\sqrt{sin\Theta }}=-\frac{\left ( 1+x^{4} \right )^{\frac{1}{4}}}{x}+c
    1. \int \frac{x^{5}}{\left ( a^{3}+x^{3} \right )^{\frac{1}{2}}}dx=\frac{1}{3}\int \left ( u^{\frac{1}{2}}-a^{3}u^{\frac{-1}{2}} \right )du=\frac{2}{9}\left ( a^{3}+x^{3} \right )^{\frac{3}{2}}-2\left ( a^{3}+x^{3} \right )^{\frac{1}{2}}+c
    2. Regards
    3. Sumit

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