Integral of (2x+5)^7 using the chain rule backwards vs. formal u-substitution approach.

We compute the integral of (2x+5)^7 using the chain rule backwards, then we show how to compute the integral of (2x+5)^7 using a formal u-substitution.

For the reverse chain rule approach, we see that the integrand is a function composition, where 2x+5 is the interior function for the composition. So we can tack on the derivative of the interior function -- that's 2 -- in the integrand and compensate with a 1/2 in front of the integral. Having the derivative of the interior function sitting right next to a power of the interior function makes it simple to recognize the chain rule. We guess the antiderivative of the integrand as 1/8*(2x+5)^8+C, and we also have a factor of 1/2 out in front to combine with the 1/8.

In the formal u-substitution approach to the integral, we still have to recognize the integrand as a function composition where the inner function is 2x+5. We let u=2x+5, and we find that du is 2dx. We transform the integral in terms of u, and we end up having to find the antiderivative of u^7, which again is 1/8*u^8. We combine with the factor of 1/2 out in front, and we get the same answer.