Mathematics
OpenStudy (anonymous):

Find integral sign 10/(4+x^2) dx using the substitution x=2 tan theta. (Give your answer in terms of x.)

OpenStudy (anonymous):

$\int\limits 10/(4+x^2) dx = 10*\int\limits1/(4+x^2) dx$ Let $x=2*\tan \theta$and $dx = 2*\sec^2\theta d\theta$ Then, $10*\int\limits 1/(4+4*\tan^2 \theta)*2*\sec^2 \theta d\theta$ Factor out a 4 in the denominator:$10*\int\limits 1/(4*(1+\tan^2\theta))*2*\sec^2\theta d\theta$ Use the trig identity $1+\tan^2\theta=\sec^2\theta$Then, $10*\int\limits (2*\sec^2\theta)/(4*\sec^2\theta) d\theta$and $10*\int\limits 1/2 d\theta=5*\int d\theta=5\theta$ Now, solve x=2*tan(theta) for theta and get $\theta=\tan^-1(x/2)$Thus, the answer is $5*\tan^-1(x/2) + c$

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