Calculus 2 integration techniques become easier when students stop asking, “What formula do I remember?” and start asking, “What structure do I see?” The method usually follows from the form of the integrand: product, composition, rational expression, radical, trig power, or improper limit.
This guide gives a practical decision process for choosing between substitution, integration by parts, trigonometric identities, trigonometric substitution, partial fractions, and numerical or improper-integral thinking.
Integration Method Decision Table
| What You See | Try This First | Example Signal |
|---|---|---|
| Inside function and its derivative | u-substitution | Integral of 2x cos(x^2) dx |
| Product of algebraic and exponential, log, inverse trig, or trig | Integration by parts | Integral of x e^x dx |
| Powers of sine and cosine | Trig identities | Integral of sin^3(x) cos^2(x) dx |
| Radicals with a^2 – x^2, a^2 + x^2, or x^2 – a^2 | Trig substitution | Integral of sqrt(9 – x^2) dx |
| Rational function | Algebra first, then partial fractions | Integral of (3x + 5)/(x^2 – x – 6) dx |
| Infinite interval or vertical asymptote | Improper integral setup | Integral from 1 to infinity of 1/x^2 dx |
Method 1: u-Substitution
Use u-substitution when part of the integrand is the derivative of another part.
Example: Integrate 2x cos(x^2) dx.
Let u = x^2, so du = 2x dx. The integral becomes integral cos(u) du = sin(u) + C = sin(x^2) + C.
Method 2: Integration by Parts
Use integration by parts for products where one factor becomes simpler when differentiated. The formula is integral u dv = uv – integral v du.
Example: Integrate x e^x dx.
Let u = x and dv = e^x dx. Then du = dx and v = e^x. The result is x e^x – integral e^x dx = x e^x – e^x + C.
Method 3: Trig Identities
Use trig identities when powers of sine, cosine, tangent, or secant block direct integration.
Example: Integrate sin^2(x) dx.
Use sin^2(x) = (1 – cos(2x))/2. Then the integral is x/2 – sin(2x)/4 + C.
Method 4: Trig Substitution
Use trig substitution when radicals match classic forms:
- a^2 – x^2 suggests x = a sin(theta)
- a^2 + x^2 suggests x = a tan(theta)
- x^2 – a^2 suggests x = a sec(theta)
Example signal: sqrt(16 – x^2) suggests x = 4 sin(theta). This transforms the radical into 4 cos(theta), which can make the integral manageable.
Method 5: Partial Fractions
Use partial fractions for rational functions after checking whether polynomial division is needed first.
Example: Integrate 1/(x^2 – 1) dx.
Factor the denominator: x^2 – 1 = (x – 1)(x + 1). Then write 1/(x^2 – 1) = A/(x – 1) + B/(x + 1). Solving gives A = 1/2 and B = -1/2. The integral is (1/2)ln|x – 1| – (1/2)ln|x + 1| + C.
Common Mistakes
Trying Integration by Parts Too Soon
If a simple substitution works, use it. Integration by parts adds complexity and can make an easy problem longer.
Ignoring Algebra Before Choosing a Method
Simplify first. Expand, factor, divide, or rewrite radicals when possible. Many “hard” integrals become basic after algebra.
Forgetting + C
In indefinite integrals, the constant of integration matters. Build the habit now because differential equations and applications rely on it.
Not Checking by Differentiating
The fastest way to catch many integration errors is to differentiate your answer and see whether it returns the original integrand.
Practice Problems
- Integrate 6x(x^2 + 4)^5 dx.
- Integrate x ln(x) dx.
- Integrate cos^2(x) dx.
- Integrate 5/(x^2 – 4) dx.
- Set up the improper integral for integral from 2 to infinity of 1/x^3 dx.
Answers and Method Notes
1. u-substitution. Let u = x^2 + 4, du = 2x dx. The answer is (1/2)(x^2 + 4)^6 + C.
2. Integration by parts. Let u = ln(x), dv = x dx. The answer is (x^2/2)ln(x) – x^2/4 + C.
3. Trig identity. cos^2(x) = (1 + cos(2x))/2, so the answer is x/2 + sin(2x)/4 + C.
4. Partial fractions. 5/(x^2 – 4) = 5/[(x – 2)(x + 2)]. The answer is (5/4)ln|x – 2| – (5/4)ln|x + 2| + C.
5. Write lim as b goes to infinity of integral from 2 to b of 1/x^3 dx.
Study Routine
For each technique, do five problems where the method is obvious, then five mixed problems where you must choose. Keep a one-page method log: integrand type, method chosen, first step, and checking step. Calculus 2 improves fastest when students practice choosing methods, not just executing them after a section title gives the method away.
The 30-Second Method Check
Before starting a long solution, spend 30 seconds testing the structure. Ask: Can I see an inside function and its derivative? Does one factor get simpler if differentiated? Is the expression rational after factoring? Is there a radical that matches a trig-substitution pattern? Is the interval improper? This short pause prevents wasted work.
For example, integral x/(x^2 + 1) dx should trigger substitution because the derivative of x^2 + 1 is 2x. But integral x arctan(x) dx should trigger integration by parts because arctan(x) becomes simpler when differentiated. The two problems both contain x, but their structures point to different methods.
Error Log Template
| Problem | Method I Chose | Better Method | What I Missed |
|---|---|---|---|
| Integral of x/(x^2+1) | Parts | Substitution | Inside function derivative was present |
| Integral of x^2/(x-1) | Partial fractions | Long division first | Numerator degree was not lower |
