Evaluate the integral using the indicated trigonometric substitution. (use c for the constant of integration.) x^3 / sqrt x^2 + 49 dx , x = 7 tan(θ)

Taking gives , so that the integral becomes







When , we have




and from here we can substitute to proceed from here.

Quick note: When we set , we are implicitly enforcing just so that the substitution can be undone later via . But note that over this domain, we automatically guarantee that , so the absolute value bars can be dropped immediately.