int frac{dx}{(a^2 + x^2)^{3/2}} is equal to | vedclass

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(B) Let $I = \int \frac{dx}{(a^2 + x^2)^{3/2}}$.Substitute $x = a 	an 	heta$,then $dx = a \sec^2 	heta \, d	heta$.Substituting these into the inte

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$\frac{x}{a^2(a^2 + x^2)^{1/2}}$

Vedclass · Answer

(B) Let $I = \int \frac{dx}{(a^2 + x^2)^{3/2}}$.Substitute $x = a 	an 	heta$,then $dx = a \sec^2 	heta \, d	heta$.Substituting these into the integral:$I = \int \frac{a \sec^2 	heta}{(a^2 + a^2 	an^2 	heta)^{3/2}} \, d	heta = \int \frac{a \sec^2 	heta}{(a^2 \sec^2 	heta)^{3/2}} \, d	heta$.$I = \int \frac{a \sec^2 	heta}{a^3 \sec^3 	heta} \, d	heta = \frac{1}{a^2} \int \frac{1}{\sec 	heta} \, d	heta$.$I = \frac{1}{a^2} \int \cos 	heta \, d	heta = \frac{1}{a^2} \sin 	heta + C$.Since $x = a 	an 	heta$,we have $	an 	heta = \frac{x}{a}$.Using the triangle identity,$\sin 	heta = \frac{x}{\sqrt{a^2 + x^2}}$.Therefore,$I = \frac{1}{a^2} \cdot \frac{x}{\sqrt{a^2 + x^2}} + C = \frac{x}{a^2(a^2 + x^2)^{1/2}} + C$.

$\int \frac{dx}{(a^2 + x^2)^{3/2}}$ is equal to

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