How do I integrate $$\int_0^\frac{\pi}{4}\frac{\sec x}{1+2\sin^2(x)}dx$$ ? Can someone help me with this? Thanks. I've tried simplifying the expression using trigonometry, and also applying properties of definite integrals but couldn't solve.
2026-04-13 02:41:43.1776048103
Question on Definite integration
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Hint $$I=\int_0^\frac{\pi}{4}\frac{\sec x}{1+2\sin^2(x)}dx$$ Substitute $u=\sin(x) \implies du=\cos(x) dx$ $$I=\int_0^\frac{\sqrt 2}{2}\frac{du}{(1+2u^2)(1-u^2)}$$ Use fraction decomposition ..