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Question Number 163368 by nurtani last updated on 06/Jan/22

Answered by Ar Brandon last updated on 06/Jan/22

$$I={\int }_0^{\frac{\pi }{2}}\frac{\frac{4}{\pi }{\sin }^{2007}x}{{\sin }^{2007}x+{\cos }^{2007}x}dx...\left(1\right)$$$$I={\int }_0^{\frac{\pi }{2}}\frac{\frac{4}{\pi }{\cos }^{2007}x}{{\sin }^{2007}x+{\cos }^{2007}x}dx...\left(2\right)$$$$\left(1\right)+\left(2\right)\Rightarrow$$$$2I=\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{2007}x+{\cos }^{2007}x}{{\sin }^{2007}x+{\cos }^{2007}x}dx=\frac{4}{\pi }{\int }_0^{\frac{\pi }{2}}dx$$$$I=\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}dx=\frac{2}{\pi }\cdot \frac{\pi }{2}=1$$

Commented by nurtani last updated on 06/Jan/22

thanks mr brandon

Commented by amin96 last updated on 06/Jan/22

$$greatsirBrandon.Integralgeniusss))$$

Commented by Ar Brandon last updated on 06/Jan/22

My pleasure, friends!

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