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Question Number 201290 by sonukgindia last updated on 03/Dec/23

Answered by Calculusboy last updated on 03/Dec/23

$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:\mathit{B}\mathit{y}\mathit{u}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{k}\mathit{i}\mathit{n}\mathit{g}\mathit{s}\mathit{r}\mathit{u}\mathit{l}\mathit{e}$$$$\mathit{I}={\int }_0^{\frac{\mathit{\pi }}{2}}\frac{{\mathit{s}\mathit{i}\mathit{n}}^{\mathit{\phi }}\left(\mathit{x}\right)}{{\mathit{s}\mathit{i}\mathit{n}}^{\mathit{\phi }}\left(\mathit{x}\right)+{\mathit{c}\mathit{o}\mathit{s}}^{\mathit{\phi }}\left(\mathit{x}\right)}\mathit{d}\mathit{x}={\int }_0^{\frac{\mathit{\pi }}{2}}\frac{{\mathit{c}\mathit{o}\mathit{s}}^{\mathit{\phi }}\left(\mathit{x}\right)}{{\mathit{c}\mathit{o}\mathit{s}}^{\mathit{\phi }}\left(\mathit{x}\right)+{\mathit{s}\mathit{i}\mathit{n}}^{\mathit{\phi }}\left(\mathit{x}\right)}\mathit{d}\mathit{x}$$$$2\mathit{I}={\int }_0^{\frac{\mathit{\pi }}{2}}\frac{{\mathit{s}\mathit{i}\mathit{n}}^{\mathit{\phi }}\left(\mathit{x}\right)+{\mathit{c}\mathit{o}\mathit{s}}^{\mathit{\phi }}\left(\mathit{x}\right)}{{\mathit{s}\mathit{i}\mathit{n}}^{\mathit{\phi }}\left(\mathit{x}\right)+{\mathit{c}\mathit{o}\mathit{s}}^{\mathit{\phi }}\left(\mathit{x}\right)}\mathit{d}\mathit{x}$$$$2\mathit{I}={\int }_0^{\frac{\mathit{\pi }}{2}}1\mathit{d}\mathit{x}$$$$2\mathit{I}=\left(\mathit{x}\right){\mid }_0^{\frac{\mathit{\pi }}{2}}+\mathit{C}$$$$2\mathit{I}=(\frac{\mathit{\pi }}{2}-0)\iff 2\mathit{I}=\frac{\mathit{\pi }}{2}$$$$\mathit{I}=\frac{\mathit{\pi }}{4}$$$$$$

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