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integration [(sin^3 X)/{(1+cos^2 X)(sqrt(1+cos^2 X+cos^4 X))}]=?

integration [(sin^3 X)/{(1+cos^2 X)(sqrt(1+cos^2 X+cos^4 X))}]=?

Grade:12th pass

1 Answers

Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
7 years ago
Solution:
Let u=cos(x),
Then the required intergral would be given by:
\int \frac{sin^{3}x}{\left ( 1+cos^{2}x \right )\left ( \sqrt{1+cos^{2}x+cos^{4}x} \right )}=\int \frac{u^{2}-1}{\left ( 1+u^{2} \right )\left ( \sqrt{1+u^{2}+u^{4}} \right )}=\int \frac{1}{\sqrt{1+u^{2}+u^{4}}}-2\int \frac{1}{\left ( 1+u^{2} \right )\sqrt{1+u^{2}+u^{4}}}The second integral can further be simpified as follows:
2\int \frac{1}{\left ( 1+u^{2} \right )\sqrt{1+u^{2}+u^{4}}}=2\int \frac{1}{\left ( 1+u^{2} \right )\sqrt{\left ( u^{2}+\frac{1}{2} \right )^{2}+\frac{3}{4}}}
Now using the standard formulae, the final result can be arrived at.
Thanks & Regards
Sumit Majumdar,
askIITians Faculty
Ph.D,IIT Delhi

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