Taking integration by parts

In this posting, we will learn another trick to compute an integral (definite or indefinite). It is called "integration by parts" because we are going to divide the integration into two parts. In your book, this is discussed in Section 7.2.

Motivating example. Let's try to compute the following indefinite integral $$\int \ln(x) dx.$$ That is, we want to classify all $F(x)$ such that $F'(x) = \ln(x).$ How would we do that? Well, if you have thought about it long enough you will realize that the following function works: $F(x) = x\ln(x) - x,$ or this with any constant added to it. Indeed, you can check that by product rule (Section 3.3), we have $$F'(x) = \ln(x) + x(1/x) - 1 = \ln(x) + 1 - 1 = \ln(x),$$ so we are good.

Q. How on earth would we be able to come up with such an antiderivative?

This is where we introduce a new trick. We really try to think about the integrand as a part of something else. Write $$\ln(x) = F'(x) = u'(x)v(x),$$ for some nicely differentiable functions $u, v.$ Note that the product consists of a derivative function and a function. If we add $u(x)v'(x),$ we have $$F'(x) + u(x)v'(x) = u'(x)v(x) + u(x)v'(x) = \frac{d}{dx} (u(x)v(x))$$ so that we have $$\ln(x) = F'(x) = \frac{d}{dx} (u(x)v(x)) - u(x)v'(x).$$ Integrating both sides, we have $$\int \ln(x) dx = u(x)v(x) - \int u(x)v'(x) dx.$$ What's such a big deal?

Philosophy. The integration by parts is a trick that once you realize $\ln(x) = u'(x)v(x)$ for smart choices of $u(x)$ and $v(x),$ you get $$\int \ln(x) dx = u(x)v(x) - \int u(x)v'(x) dx$$ so that the computation of the left-hand side reduces to the computation of $\int u(x)v'(x) dx.$ If the last integral is easy to compute, then you win!

Exercise. Choose $u'(x) = 1$ and $v(x) = \ln(x)$ to finish the computation above.

Exercise. Compute $\int t \sin(t) dt.$

Exercise (Hard). Compute $\int \sin^{2}(t) dt.$

Exercise. Compute $\int_{e}^{3} x \ln(x) dx.$

Exercise. Compute $\int x^{2} \ln(x^{3}) dx.$

Exercise. For any positive integer $n,$ compute $\int x^{n} \ln(x) dx.$ (Hint: compute $\int x^{n} \ln(x^{n+1}) dx$ first using the substitution $u = x^{n+1},$ and use $(n+1) \ln(x) = \ln(x^{n+1}).$)

Exercise (Hard). Compute $\int e^{t}\cos(2t) dt.$

Exercise. Do #1 (a) on Exam 1 (Winter 2019).

Exercise. Do #3 on Exam 1 (Winter 2019).

Exercise. Do #1 (c) on Exam 1 (Fall 2018).