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Exercise 4.2 · Q34

Q.Evaluate: ∫1/21ecos⁡−1x sin⁡−1x1−x2 dx\int_{1/\sqrt2}^1 \dfrac{e^{\cos^{-1}x}\,\sin^{-1}x}{\sqrt{1-x^2}}\,dx

Maharashtra MsbshseTextbookSubjectiveImportance★★★★★
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Let θ=sin⁡−1x\theta=\sin^{-1}x, so cos⁡−1x=π2−θ\cos^{-1}x=\frac\pi2-\theta and dθ=dx1−x2d\theta=\dfrac{dx}{\sqrt{1-x^2}}; limits x=12→θ=π4x=\frac1{\sqrt2}\to\theta=\frac\pi4, x=1→θ=π2x=1\to\theta=\frac\pi2.

∫π/4π/2eπ/2−θ θ dθ=eπ/2∫π/4π/2θe−θ dθ.\int_{\pi/4}^{\pi/2} e^{\pi/2-\theta}\,\theta\,d\theta=e^{\pi/2}\int_{\pi/4}^{\pi/2}\theta e^{-\theta}\,d\theta.

Using ∫θe−θdθ=−(θ+1)e−θ\int\theta e^{-\theta}d\theta=-(\theta+1)e^{-\theta} (by parts): …

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