Tutorial on Integration Techniques

Integration Techniques

Trigonometric Substitution

• Used when integrand contains $\sqrt{a^2 - x^2}$, $\sqrt{x^2 + a^2}$, or $\sqrt{x^2 - a^2}$
• Key substitutions:
  • $\sqrt{a^2 - x^2} \Rightarrow x = a\sin\theta$
  • $\sqrt{x^2 + a^2} \Rightarrow x = a\tan\theta$
  • $\sqrt{x^2 - a^2} \Rightarrow x = a\sec\theta$
• Example: For $\int \frac{dx}{x^2\sqrt{4-x^2}}$, let $x = 2\sin\theta$
• Useful integrals:
  • $\int \csc\alpha, d\alpha = -\ln|\csc\alpha + \cot\alpha| + C$
  • $\int \sec\alpha, d\alpha = \ln|\sec\alpha + \tan\alpha| + C$

Integration by Parts

• Formula: $\int u,dv = uv - \int v,du$
• Choose $u$ using LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

Partial Fractions

• Decompose $\frac{P(x)}{Q(x)}$ where degree$(P) <$ degree$(Q)$
• For distinct linear factors $(x-a)(x-b)$: $$\frac{A}{x-a} + \frac{B}{x-b}$$
• For repeated factors $(x-a)^2$: $$\frac{A}{x-a} + \frac{B}{(x-a)^2}$$
• For irreducible quadratics $x^2 + a^2$: $$\frac{Ax+B}{x^2+a^2}$$

Definite Integrals

• Evaluate antiderivative at bounds: $\int_a^b f(x) dx = F(b) - F(a)$
• Absolute values: Split integral at points where expression changes sign
  • Example: $\int_0^4 |x^2-2|,dx = \int_0^{\sqrt{2}} (2-x^2)dx + \int_{\sqrt{2}}^4 (x^2-2)dx$
• Properties:
  • Symmetry: $\int_{-a}^a f(x)dx = 0$ if $f$ odd, $= 2\int_0^a f(x)dx$ if $f$ even
  • Additivity: $\int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx$