Integration by Parts | Integration Techniques

What You'll Learn

1. The Formula

Theorem 1.1: Integration by Parts

\int u\,dv = uv - \int v\,du

This is the reverse of the product rule for differentiation.

Choose $u$ in this priority order:

Example 2.1: Polynomial times Exponential

Find: $\int xe^x\,dx$

Solution:

Let $u = x$ , $dv = e^x\,dx$

Then $du = dx$ , $v = e^x$

\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1) + C

Example 2.2: Logarithmic Function

Find: $\int \ln x\,dx$

Solution:

Let $u = \ln x$ , $dv = dx$

Then $du = \frac{1}{x}dx$ , $v = x$

\int \ln x\,dx = x\ln x - \int x \cdot \frac{1}{x}\,dx = x\ln x - x + C

Example 2.3: Repeated Integration by Parts

Find: $\int x^2e^x\,dx$

Solution:

First application: Let $u = x^2$ , $dv = e^x\,dx$

\int x^2e^x\,dx = x^2e^x - \int 2xe^x\,dx

Second application: For $\int 2xe^x\,dx$ , let $u = 2x$ , $dv = e^x\,dx$

\int 2xe^x\,dx = 2xe^x - 2e^x + C

Combine:

\int x^2e^x\,dx = x^2e^x - 2xe^x + 2e^x + C = e^x(x^2-2x+2) + C

Example 2.4: Reduction Formula

Find: $\int e^x\sin x\,dx$

Solution:

Apply integration by parts twice:

I = \int e^x\sin x\,dx = e^x\sin x - \int e^x\cos x\,dx

= e^x\sin x - (e^x\cos x + \int e^x\sin x\,dx)

Notice $\int e^x\sin x\,dx$ appears again!

I = e^x\sin x - e^x\cos x - I

2I = e^x(\sin x - \cos x)

I = \frac{e^x(\sin x - \cos x)}{2} + C

Reinforce integration techniques through guided practice