Integration by parts

Statement

Formal manipulation version for indefinite integration in function notation

Suppose $F$ and $g$ are continuous functions such that $F$ is a differentiable function and we want to integrate $F(x)g(x)$. Suppose $G$ is an antiderivative for $g$. Then, we have:

$\int F(x)g(x)\,dx=F(x)G(x)-\int F'(x)G(x)\,dx$

Verbally, the integral of a product of two functions is the first function times the integral of the second function minus the integral of the derivative of the first function times the second function.

Formal manipulation version in variable transformation notation

Suppose $u,v$ are variables denoting functions of $x$. Then, we have:

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

More explicitly:

$\int u{\frac {dv}{dx}}\,dx=uv-\int v{\frac {du}{dx}}\,dx$

Compared with the notation of the preceding version, $u=F(x)$, $dv/dx=g(x)$, and $v=G(x)$.

Formal manipulation version for definite integration in function notation

Suppose $F$ and $g$ are continuous functions on a closed interval $[a,b]$ such that $F$ is a differentiable function on the open interval $(a,b)$ and we want to integrate $F(x)g(x)$. Suppose $G$ is an antiderivative for $g$. Then, we have:

$\int _{a}^{b}F(x)g(x)\,dx=[F(x)G(x)]_{a}^{b}-\int _{a}^{b}F'(x)G(x)\,dx$

Verbally, the integral of a product of two functions is the first function times the integral of the second function minus the integral of the derivative of the first function times the second function.

Key observations

Equivalence of integration problems

Integration by parts tells us the following:

PUSH DOWN ONE PART, PUSH UP THE OTHER PART OF THE PRODUCT: Integrating a product of two functions is equivalent to integrating a new product where we have differentiated one of the functions and integrating the other.

Repeated use of integration by parts and the circular trap

Integration by parts can be used multiple times, i.e., the new integration that we obtain from an application of integration by parts can again be subjected to integration by parts. However, we need to make sure that we avoid the circular trap:

AVOID THE CIRCULAR TRAP: When using integration by parts a second time, make sure you don't choose as the part to integrate the thing you got by differentiating the part to differentiate from the original product. Otherwise, you get in a circular trap and don't get any new information. The most typical application of integration by parts a second time is if you choose to differentiate again the expression that you already obtained through differentiation the first time

The typical repeated application of integration by parts looks like:

Failed to parse (unknown function "\leadsto"): {\displaystyle F(x)g(x) \leadsto F'(x)\int g(x) \, dx \leadsto F''(x) \int int g(x) \, dx \, dx \leadsto \dots}