Strategy for Integrals using Integration by Parts I = ∫ u dv = uv ˗ ∫ v du

In a Nut Shell: There are three common types of integrals where integration by parts is
the recommended procedure. It is best to identify these types before applying integration
by parts. The table below lists these types along with examples.

Strategy for Type 1

The integral, I, has the following form: ∫ f(x) g(x) dx. Here you know the integrals for both f(x) and g(x). So you need to pick u and dv such that it simplifies the integral ∫ v du compared with the original integral ∫ f(x) g(x) dx. Example: I = ∫ x cos x dx Here you know both integrals for x and for cos x. Pick u = x and dv = cos x dx Then du = dx and v = sin x so I = x sinx ˗ ∫ sin x dx Finally I = x sin x + cos x + C (result) Note: If you were to pick u = cos x and dv = x dx the next integral Then du = ˗ sin x dx and v = (x^2 / 2) and the second integral would be more complicated. i.e. ∫ ( (x^2 / 2) sin x dx than the one you started out with.

Strategy for Type 2

The integral, I, has the following form: ∫ f(x) g(x) dx. Here suppose you only know the integral for one of the two functions, f(x) and g(x). So you need to pick u for the function where you don't know its integral. Example: I = ∫ x ln x dx Here you don't know the integral for ln x. Therefore pick u = ln x and dv = x Then du = (1/x) dx and v = (x^2 / 2) so I = (x^2 / 2) ln x ˗ ∫ (x/2) dx Finally I = (x^2 / 2) ln x ˗ x^2 /4 + C (result)

Click here to continue with Type 3.

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