Question:

By using the properties of definite integrals, evaluate the integral: \(∫^{\frac{π}{2}}_0\frac{cos^5xdx}{sin^5x+cos^5x}\)

Updated On: Oct 7, 2023
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Solution and Explanation

The correct answer is:\(I=\frac{π}{4}\)
Let \(I=∫^{\frac{π}{2}}_0\frac{cos^5xdx}{sin^5x+cos^5x}......(1)\)
\(⇒I=∫^{\frac{π}{2}}_0\frac{cos^5(\frac{π}{2}-x)}{sin^5(\frac{π}{2}-x)+cos^5(\frac{π}{2}-x)}dx\,\,\,\, (∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)\)
\(⇒I=∫^{\frac{π}{2}}_0\frac{sin^5x}{sin^5x+cos^5x}dx...(2)\)
Adding(1)and(2),we obtain
\(2I=∫^{\frac{π}{2}}_0\frac{sin^5x+cos^5x}{sin^5x+cos^5x}dx\)
\(⇒2I=∫^{\frac{π}{2}}_01.dx\)
\(⇒2I=[x]^{\frac{π}{2}}_0\)
\(⇒2I=\frac{π}{2}\)
\(⇒I=\frac{π}{4}\)
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