Trigonometric integral sin(x)*(sec(x))^3: change to sine and cosine, guess the chain rule backwards.

We are given the trigonometric integral sin(x)*(sec(x))^3 and we change to sine and cosine to simplify the integrand. Immediately we arrive at a power of cosine multiplied by sine, and we guess the chain rule backwards by setting up the derivative of the interior function next to a power of the inner function. We adjust by supplying a factor of -1 inside the integral and compensate in front of the integral.

We quickly obtain the antiderivative and express it in two different ways, one in terms of cosine and the other in terms of secant.