Use the indicated substitution to evaluate the integral

1

(integral) dx/(x^2)sqrt{(x^2)+4), x=2tan(θ)

1/2

first got the derivative of x=2tan(θ)-->dx=2sec^2 (θ) dθ

then I evaluated the denominator, which becomes 8tan^2(θ)secθ

1

(integral) dx/(x^2)sqrt{(x^2)+4) =

1/2

integral 1,1/2: 2sec^2(θ)dθ/8tan^2(θ)sec(θ) =

(1/4) integral 1,1/2: 1dθ/cos(θ)(sin(θ)/cos(θ))(tan(θ))

(1/4) integral 1,1/2: 1/sin(θ)tan(θ) dθ

(1/4 integral 1,1/2: csc(θ) dθ) x (1/4 integral 1,1/2: cot(θ) dθ)

1

(1/4)(ln|csc(θ)-cot(θ)|) x (ln|sin(θ)|) + C

1/2

Thanks!