Integration by Parts

Definition

Integration by parts is based on the product rule for differentiation. Since $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$, rearranging and integrating gives

$$\int f^{\prime }(x)g(x)dx=f(x)g(x)-\int f(x)g^{\prime }(x)dx.$$

Intuition

Integration by parts is the product rule run backwards: it trades one integral for another by shifting a derivative from one factor to the other. The goal is to choose $u$ and $dv$ so that the new integral $\int vdu$ is simpler than the original. A useful mnemonic for choosing $u$ is LIATE (Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential) — pick $u$ from the category that appears earliest in the list.

Formal Description

Setting $u=g(x)$, $dv=f^{\prime }(x)dx$, $du=g^{\prime }(x)dx$, $v=f(x)$, the formula becomes

$$\int udv=uv-\int vdu.$$

The key idea is that $\int vdu$ should be simpler than $\int udv$.

For definite integrals:

$${\int }_a^{b}udv=uv{|}_a^b-{\int }_a^{b}vdu.$$

Example. $\int x{e}^{x}dx$. Let $u=x$, $dv=e^{x}dx$, so $du=dx$, $v=e^x$:

$$\int x{e}^{x}dx=x{e}^x-\int e^{x}dx=x{e}^x-e^x+C=e^x(x-1)+C.$$

Applications

• Integrals of the form $\int x^n{e}^{x}dx$, $\int x^n\sin xdx$, $\int x^n\cos xdx$ (reduce the power of $x$ by repeated application).
• Integrals involving $\ln x$ or inverse trigonometric functions, where these are chosen as $u$.
• Deriving reduction formulas for $\int {\sin }^{n}xdx$, $\int {\cos }^{n}xdx$, and similar families.

Trade-offs

• The technique requires a good choice of $u$ and $dv$; a poor choice yields a more complicated integral.
• Some integrals require applying the formula twice (or more), and occasionally the original integral reappears on the right — it can then be solved algebraically rather than integrated again.
• Integration by parts does not apply when the integrand cannot be written as a product of two functions in a useful way.
• For definite integrals, both the boundary term $uv{|}_a^b$ and the remaining integral must be evaluated; errors in either part invalidate the result.

Links

• Product Rule
• Fundamental Theorem of Calculus
• Integrals