Most functions resist direct antidifferentiation. The integral of ex2 has no elementary formula. Even innocuous-looking expressions like 1+x3 lack closed-form antiderivatives.
Integration techniques transform difficult integrals into tractable ones. Substitution reverses the chain rule. Integration by parts reverses the product rule. Partial fractions decompose rational functions. Trigonometric methods handle roots and powers.
No single algorithm covers all cases—unlike differentiation, which follows systematic rules. Success requires recognizing which technique applies and executing it correctly. This pattern-matching skill develops through practice.
Why Techniques Are Needed
Differentiation follows mechanical rules: the derivative of any elementary function can be computed by applying chain, product, and quotient rules systematically.
Integration has no such universal algorithm. Some elementary functions have no elementary antiderivative. Others have antiderivatives that are difficult to find without insight.
Techniques bridge this gap. They transform integrands into forms matching known formulas. The transformation might involve changing variables, splitting products, or decomposing fractions—whatever reveals the underlying structure.
Substitution (u-Substitution)
Substitution reverses the chain rule. If the integrand contains a function and its derivative, substitution simplifies.
Method: Let u=g(x), so du=g′(x)dx. Replace all x-expressions with u-expressions and integrate.
Example:
∫2xcos(x2)dx
Let u=x2, so du=2xdx:
=∫cosudu=sinu+C=sin(x2)+C
For definite integrals, convert the limits: when x=a, u=g(a); when x=b, u=g(b).
Integration by Parts
Integration by parts reverses the product rule:
∫udv=uv−∫vdu
Method: Identify factors u and dv in the integrand. Differentiate u to get du; integrate dv to get v. Apply the formula.
Example:
∫xexdx
Let u=x and dv=exdx. Then du=dx and v=ex:
=xex−∫exdx=xex−ex+C=ex(x−1)+C
Choosing $u$: LIATE guides selection—Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Earlier types typically make better choices for u.
Trigonometric Integrals
Integrals involving powers of sine and cosine require strategic use of identities.
Odd power of sine: Save one sinx, convert remaining sin2x=1−cos2x, substitute u=cosx.
Odd power of cosine: Save one cosx, convert remaining cos2x=1−sin2x, substitute u=sinx.
Both powers even: Use half-angle identities:
sin2x=21−cos2xcos2x=21+cos2x
Example:
∫sin3xdx=∫sinx(1−cos2x)dx
Let u=cosx:
=−∫(1−u2)du=−u+3u3+C=−cosx+3cos3x+C
Trigonometric Substitution
Square roots of quadratics suggest trigonometric substitutions.
For $\sqrt{a^2 - x^2}$: Let x=asinθ, so a2−x2=acosθ
For $\sqrt{a^2 + x^2}$: Let x=atanθ, so a2+x2=asecθ
For $\sqrt{x^2 - a^2}$: Let x=asecθ, so x2−a2=atanθ
Example:
∫1−x21dx
Let x=sinθ, so dx=cosθdθ and 1−x2=cosθ:
=∫cosθcosθdθ=∫dθ=θ+C=arcsinx+C
Partial Fractions
Rational functions—polynomials divided by polynomials—decompose into simpler fractions.
Method: Factor the denominator. Write the fraction as a sum of terms with linear or irreducible quadratic denominators. Solve for coefficients. Integrate each term.
Example:
∫x2−11dx=∫(x−1)(x+1)1dx
Decompose:
(x−1)(x+1)1=x−1A+x+1B
Solving gives A=1/2, B=−1/2:
=21∫x−11dx−21∫x+11dx=21ln∣x−1∣−21ln∣x+1∣+C
Choosing the Right Technique
Pattern recognition guides technique selection.
Substitution: Look for a function paired with its derivative. Expressions like f(g(x))⋅g′(x) signal substitution with u=g(x).
Parts: Products of different function types—polynomial times exponential, polynomial times trigonometric, logarithm times polynomial.
Trigonometric integrals: Powers of sinx and cosx, products of trigonometric functions.
Trigonometric substitution: Square roots of a2−x2, a2+x2, or x2−a2.
Partial fractions: Rational functions with factorable denominators.
Multiple techniques often combine. A problem might require substitution followed by parts, or partial fractions followed by a trigonometric integral.