Euler's Method

Advanced Integration Techniques

By the end of this lesson, you will be able to apply integration by parts, use partial fractions, evaluate improper integrals, and apply Euler's method.

Integration by Parts

The formula:

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

This is the "undo" of the product rule for differentiation.

LIATE rule for choosing $u$: choose $u$ as the first in this order:

• Logarithmic functions
• Inverse trig functions
• Algebraic (polynomial)
• Trigonometric functions
• Exponential functions

(What's left over becomes $dv$.)

Example 1: $\displaystyle\int x e^x, dx$

Let $u = x$ (algebraic), $dv = e^x, dx$

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

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

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

Let $u = \ln x$ (logarithmic — LIATE says L first), $dv = x, dx$

$du = \dfrac{1}{x}, dx$, $v = \dfrac{x^2}{2}$

$= \dfrac{x^2}{2}\ln x - \displaystyle\int \dfrac{x^2}{2} \cdot \dfrac{1}{x}, dx = \dfrac{x^2}{2}\ln x - \dfrac{1}{2}\int x, dx = \dfrac{x^2}{2}\ln x - \dfrac{x^2}{4} + C$

Example 3 (repeated): $\displaystyle\int e^x \sin x, dx$

Let $u = \sin x$, $dv = e^x, dx$ → $du = \cos x, dx$, $v = e^x$

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

Apply IBP again: $u = \cos x$, $dv = e^x, dx$

$= e^x\sin x - \left[e^x\cos x + \displaystyle\int e^x\sin x, dx\right]$

Let $I = \displaystyle\int e^x\sin x, dx$:

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

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

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

Partial Fraction Decomposition

Purpose: integrate rational functions $\dfrac{P(x)}{Q(x)}$ by rewriting as simpler fractions.

When to use: numerator degree < denominator degree; denominator factors.

Case 1: Distinct linear factors

$\dfrac{3x + 1}{(x-1)(x+2)} = \dfrac{A}{x-1} + \dfrac{B}{x+2}$

Multiply both sides by $(x-1)(x+2)$: $3x + 1 = A(x+2) + B(x-1)$

$x = 1$: $4 = 3A \Rightarrow A = 4/3$

$x = -2$: $-5 = -3B \Rightarrow B = 5/3$

$\displaystyle\int \dfrac{3x+1}{(x-1)(x+2)}, dx = \dfrac{4}{3}\ln|x-1| + \dfrac{5}{3}\ln|x+2| + C$