Grade 12th passIntegral Calculus

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

mamillapelly bhagath

12 Years agoGrade 12th pass

1 Answer

Sumit Majumdar

12 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:

$2\int \frac{1}{\left ( 1+u^{2} \right )\sqrt{1+u^{2}+u^{4}}}=2\int \frac{1}{\left ( 1+u^{2} \right )\sqrt{\left ( u^{2}+\frac{1}{2} \right )^{2}+\frac{3}{4}}}$