8.1 Integration by Parts

8.1 Integration by Parts

Introduction to Integration by Parts

Integration by parts is a fundamental technique in calculus for integrating products of functions. It is derived from the product rule for differentiation and is especially useful when direct integration is difficult.

• Product Rule for Differentiation: If u(x) and v(x) are differentiable functions, then:

$\frac{d}{dx}\left[u(x)v(x)\right] = u(x)\frac{dv(x)}{dx} + v(x)\frac{du(x)}{dx}$

• By integrating both sides and rearranging, we obtain the formula for integration by parts:

$\int u(x)\,dv(x) = u(x)v(x) - \int v(x)\,du(x)$

General Form and Strategy

The general form of integration by parts is:

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

• Strategy: Choose u and dv such that differentiating u simplifies the integral, and integrating dv is straightforward.

• The goal is to transform a difficult integral into a simpler one.

• Sometimes, it may be necessary to apply integration by parts more than once.

Examples of Integration by Parts

Example 1: $\int x \cos x\,dx$

• Let $u = x$, $dv = \cos x\,dx$

• Then $du = dx$, $v = \sin x$

• Applying the formula:

$\int x \cos x\,dx = x \sin x - \int \sin x\,dx = x \sin x + \cos x + C$

Example 2: $\int \ln x\,dx$

• Let $u = \ln x$, $dv = dx$

• Then $du = \frac{1}{x}dx$, $v = x$

• Applying the formula:

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

Example 4: $\int e^x \cos x\,dx$

• This integral requires using integration by parts twice.

• Let $u = e^x$, $dv = \cos x\,dx$

• Then $du = e^x dx$, $v = \sin x$

• Applying the formula:

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

• Apply integration by parts again to $\int e^x \sin x\,dx$.

Reduction Formulas and Advanced Applications

Integration by parts can be used to derive reduction formulas, which express an integral in terms of a similar integral with a lower power or simpler form.

• For example, for $\int \cos^n x\,dx$:

$\int \cos^n x\,dx = \frac{1}{n} \sin x \cos^{n-1} x + \frac{n-1}{n} \int \cos^{n-2} x\,dx$

• Reduction formulas are useful for evaluating integrals involving powers of trigonometric functions.

Summary Table: Integration by Parts Steps

Step	Description
1. Identify $u$ and $dv$	Choose $u$ to simplify upon differentiation; $dv$ should be easy to integrate.
2. Compute $du$ and $v$	Differentiate $u$ to get $du$; integrate $dv$ to get $v$.
3. Apply the formula	Use $\int u\,dv = uv - \int v\,du$.
4. Simplify	Evaluate the remaining integral and combine terms.

Additional info: Integration by parts is especially useful for integrals involving products of polynomials, exponentials, logarithms, and trigonometric functions. The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) is a common heuristic for choosing u.