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Question Number 189293 by Rupesh123 last updated on 14/Mar/23

Answered by Sutrisno last updated on 14/Mar/23

$${\int }_0^{\frac{\pi }{2}}\frac{{sin}^{2}x}{\frac{1}{{sin}^{2}x}+\frac{{cos}^{2}x}{{sin}^{2}x}}dx$$$${\int }_0^{\frac{\pi }{2}}\frac{{sin}^{4}x}{1+{cos}^{2}x}dx$$$${\int }_0^{\frac{\pi }{2}}\frac{{(1-{cos}^{2}x)}^2}{1+{cos}^{2}x}dx$$$${\int }_0^{\frac{\pi }{2}}\frac{{cos}^{4}x-2{cos}^{2}x+1}{1+{cos}^{2}x}dx$$$${\int }_0^{\frac{\pi }{2}}{cos}^{2}x-3+\frac{4}{{cos}^{2}x+1}dx$$$${\int }_0^{\frac{\pi }{2}}\frac{1}{2}cos2x-\frac{5}{2}+\frac{4}{{cos}^{2}x+1}dx$$$$\frac{1}{4}sin2x-\frac{5}{2}x+2\sqrt{2}{tan}^{-1}\left(\frac{tanx}{\sqrt{2}}\right){\mid }_0^{\frac{\pi }{2}}$$$$(0-\frac{5\pi }{4}+2\sqrt{2}.\frac{\pi }{2})-0$$$$\frac{\pi }{4}(4\sqrt{2}-5)$$$$\left(\frac{4(5+4\sqrt{2})}{7}\right).\frac{\pi }{4}(4\sqrt{2}-5)$$$$\frac{\pi }{7}(32-25)$$$$\pi$$$$$$$$$$$$$$$$$$$$$$$$$$

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