Evaluate: ∫π/4π/2 dfraccosθ[cos(θ)/(2)+sin(θ)/(2)]³dθ

Miscellaneous Exercise 4 · Q62

\int_{\pi/4}^{\pi/2} \dfrac{\cos\theta}{\left[\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\right]^3}\,d\theta

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$\big[\cos\tfrac\theta2+\sin\tfrac\theta2\big]^2=1+2\sin\tfrac\theta2\cos\tfrac\theta2=1+\sin\theta$ , and since $\theta/2\in[\pi/8,\pi/4]$ both terms are positive, so $\big[\cos\tfrac\theta2+\sin\tfrac\theta2\big]^3=(1+\sin\theta)^{3/2}$ .

Let $u=1+\sin\theta,\ du=\cos\theta\,d\theta$ ; limits $\theta=\pi/4\to u=1+\tfrac{\sqrt2}2$ , $\theta=\pi/2\to u=2$ .

$\int u^{-3/2}\,du=-2u^{-1/2}=-\frac2{\sqrt u}.$ …

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Q.Evaluate: $\int_{\pi/4}^{\pi/2} \dfrac{\cos\theta}{\left[\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\right]^3}\,d\theta$

Concept understanding — Evaluation of Definite Integrals by Substitution

Q.Evaluate: ∫π/4π/2cos⁡θ[cos⁡θ2+sin⁡θ2]3 dθ\int_{\pi/4}^{\pi/2} \dfrac{\cos\theta}{\left[\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\right]^3}\,d\theta∫π/4π/2​[cos2θ​+sin2θ​]3cosθ​dθ

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Q.Evaluate: $\int_{\pi/4}^{\pi/2} \dfrac{\cos\theta}{\left[\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\right]^3}\,d\theta$

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