Evaluate the integral: int sec^2 theta (sec t | vedclass

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Question

(C) Let $t = \sec 	heta + 	an 	heta$. Then $dt = (\sec 	heta 	an 	heta + \sec^2 	heta) d	heta = \sec 	heta (	an 	heta + \sec 	heta) d	het

Vedclass · Accepted Answer

$\frac{(\sec 	heta + 	an 	heta)}{3} [2 + 	an 	heta (\sec 	heta + 	an 	heta)] + C$

Vedclass · Answer

(C) Let $t = \sec 	heta + 	an 	heta$. Then $dt = (\sec 	heta 	an 	heta + \sec^2 	heta) d	heta = \sec 	heta (	an 	heta + \sec 	heta) d	heta$. From $t = \sec 	heta + 	an 	heta$,we have $\sec 	heta - 	an 	heta = \frac{1}{t}$. Adding the two equations: $2 \sec 	heta = t + \frac{1}{t} \implies \sec 	heta = \frac{t^2 + 1}{2t}$. Subtracting the two equations: $2 	an 	heta = t - \frac{1}{t} \implies 	an 	heta = \frac{t^2 - 1}{2t}$. Now,$\sec^2 	heta = \sec 	heta \cdot \sec 	heta = \frac{t^2 + 1}{2t} \cdot \sec 	heta$. Substituting into the integral: $\int \sec^2 	heta (\sec 	heta + 	an 	heta)^2 d	heta = \int \sec 	heta \cdot \sec 	heta \cdot t^2 d	heta$. Since $dt = \sec 	heta \cdot t d	heta$,we have $d	heta = \frac{dt}{t \sec 	heta}$. Integral becomes $\int \sec 	heta \cdot t^2 \cdot \frac{dt}{t} = \int \sec 	heta \cdot t dt = \int \frac{t^2 + 1}{2t} \cdot t dt = \frac{1}{2} \int (t^2 + 1) dt = \frac{1}{2} (\frac{t^3}{3} + t) + C = \frac{t}{6} (t^2 + 3) + C$. Substituting $t = \sec 	heta + 	an 	heta$: $\frac{(\sec 	heta + 	an 	heta)}{6} ((\sec 	heta + 	an 	heta)^2 + 3) + C$. Expanding $(\sec 	heta + 	an 	heta)^2 = \sec^2 	heta + 	an^2 	heta + 2 \sec 	heta 	an 	heta = 1 + 2 	an^2 	heta + 2 \sec 	heta 	an 	heta$. Result: $\frac{(\sec 	heta + 	an 	heta)}{6} (1 + 2 	an^2 	heta + 2 \sec 	heta 	an 	heta + 3) = \frac{(\sec 	heta + 	an 	heta)}{6} (4 + 2 	an^2 	heta + 2 \sec 	heta 	an 	heta) = \frac{(\sec 	heta + 	an 	heta)}{3} (2 + 	an^2 	heta + \sec 	heta 	an 	heta) = \frac{(\sec 	heta + 	an 	heta)}{3} [2 + 	an 	heta (	an 	heta + \sec 	heta)] + C$.

Evaluate the integral: $\int \sec^2 \theta (\sec \theta + \tan \theta)^2 d\theta$

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