Question:
By using the properties of definite integrals, evaluate the integral: \(∫^{2π}_0 cos^5 xdx\)
By using the properties of definite integrals, evaluate the integral: \(∫^{2π}_0 cos^5 xdx\)
Updated On: Oct 7, 2023
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Solution and Explanation
Let I=\(∫^{2π}_0 cos^5 xdx...(1)\)
\(cos^5(2π-x)=cos^5x\)
It is known that,
\(∫^{2a}_0ƒ(x)dx=2∫^a_0ƒ(x)dx,if\, ƒ(2a-x)=ƒ(x)\)
\(=0\,\, if\,\, ƒ(2a-x)=-ƒ(x)\)
\(∴I=2∫^π_0cos^5 xdx \)
\(⇒I=2(0)=0 \,\,\,\,[cos^5(π-x)=-cos^5x]\)
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