Question

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)

Accepted Answer

Recognize that the integral has the form $$\int \frac{dy}{(1 - y^2$)^{5/2}$$}$$, which suggests a trigonometric substitution because of the 1$$ - $$y^2$$ term under the power. Use the substitution $$y = \sin 	heta$$, which implies $$dy = \cos 	heta \, d	heta. $$This substitution transforms the integral into one involving powers of $$\cos 	heta. $$Rewrite the integral limits in terms of $$	heta$$: when $$y = 0$$, $$	heta = \arcsin(0) = 0$$; when $$y = \frac{\sqrt{3}}{2}$$, $$	heta = \arcsin\left(\frac{\sqrt{3}}{2}ight) = \frac{\pi}{3}. $$Substitute into the integral: replace $$dy$$ with $$\cos 	heta \, d	heta$$ and $$1 - y^2$ with $$\$$cos^2$$ 	heta$$, so the integral becomes $$\$$int_0$$^{\pi/3} \frac{\cos 	heta \, d	heta}{(\$$cos^2$$ \$$theta)^{5/2}$$} = \$$int_0$$^{\pi/3} \frac{\cos 	heta \, d	heta}{\$$cos^5$$ 	heta} = \$$int_0$$^{\pi/3} \$$sec^4$$ 	heta \, d	heta$$. Apply the reduction formula for $$\int \sec^n 	heta \, d	heta$$ with n=4$ to evaluate the integral $$\$$int_0$$^{\pi/3} \$$sec^4$$ 	heta \, d	heta$. This formula helps express the integral in terms of lower powers of secant and tangent functions.

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)

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Concetti chiave

Trigonometric Substitution

Reduction Formulas

Definite Integration with Limits