Flag Integral Calculus> integration [(sin^3 X)/{(1+cos^2 X)(sqrt(...
question mark

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

mamillapelly bhagath , 11 Years ago
Grade 12th pass
anser 1 Answers
Sumit Majumdar

Last Activity: 11 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

star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments