I need a hint for evaluating $$\int_{0}^{\pi / 2} \ln \left(1+3 \sin ^{2} x\right) d x$$ I've stumbled upon this integral which is from a calculus textbook of mine and I have no idea how to solve it. I don't see how a substitution would do the work here. Any hints?
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4Does this answer your question? Integral $\int_0^{\pi/2} \ln(1+\alpha\sin^2 x), dx=\pi \ln \frac{1+\sqrt{1+\alpha}}{2}$ – Gareth Ma Jul 08 '21 at 23:11
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Consider the parametric integral with $a \geq 0$ : $$ F(a)=\int_{0}^{\pi / 2} \ln \left(1+a \sin ^{2} x\right) d x $$ Differentiating both sides with respect to $a$ yields $$ \begin{aligned} F^{\prime}(a) &=\int_{0}^{\pi / 2} \frac{\partial}{\partial a} \ln \left(1+a \sin ^{2} x\right) d x \\ &=\int_{0}^{\pi / 2} \frac{\sin ^{2} x}{1+a \sin ^{2} x} d x \end{aligned} $$ Also, note that $F(0)=0$ and solve for $F(a)$ by integrating both sides of what you've ended up with for $F^{\prime}(a)$ from $0$ to $a$. Can you take it from here?
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