Integration Techniques

U-Substitution (Review)

$\int f(g(x))g'(x)\,dx = \int f(u)\,du$ where $u = g(x)$.

Integration by Parts

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

LIATE priority for choosing $u$: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

Common patterns:

• $\int x e^x\,dx$: $u = x$, $dv = e^x\,dx$.
• $\int x^2 \sin x\,dx$: apply by parts twice.
• $\int \ln x\,dx$: $u = \ln x$, $dv = dx$.

Partial Fractions

For $\int \frac{P(x)}{Q(x)}\,dx$ where degree of $P <$ degree of $Q$:

1. Factor the denominator.
2. Decompose into partial fractions.
3. Integrate each fraction.

Example: $\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$.

Trigonometric Integrals

• $\int \sin^m x \cos^n x\,dx$: use identities or substitution.
• Useful identities:
  • $\sin^2 x = \frac{1 - \cos 2x}{2}$
  • $\cos^2 x = \frac{1 + \cos 2x}{2}$

Improper Integrals (BC-specific)

$\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$

$\int_a^b f(x)\,dx \text{ where } f \text{ has a vertical asymptote}$

Convergent if the limit exists and is finite; divergent otherwise.

Problem: Evaluate $\int x\ln x\,dx$.

Solution: By parts: $u = \ln x$, $dv = x\,dx$. So $du = dx/x$, $v = x^2/2$.

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

• U-substitution for composite functions.
• Integration by parts: $\int u\,dv = uv - \int v\,du$ (use LIATE).
• Partial fractions: decompose rational functions.
• Improper integrals: take limits; check convergence.