1. Why Substitution Is Needed
Some integrals become much simpler when a part of the expression is treated as a new variable. Substitution helps turn a difficult integrand into an easier one. In personal-notes style: substitution is like changing the viewpoint of the integral to make it look simpler.
2. Idea Behind Substitution
Substitution works by reversing the chain rule from differentiation. When a function has an inside expression, we replace that inside with a new variable to simplify the integral.
2.1. Main Step
Choose a part of the integrand (often the inside function) and set:
u = g(x)
Then compute du to replace dx.
3. How to Perform Substitution
The process is straightforward once the right substitution is chosen.
3.1. Step-by-Step Method
- Pick a substitution u = g(x).
- Compute du = g'(x) dx.
- Rewrite the entire integral in terms of u and du.
- Integrate with respect to u.
- Substitute back the original variable.
4. Choosing the Right Substitution
The best substitution usually simplifies the integrand. Some common patterns are helpful to remember.
4.1. Useful Patterns
- Look for an inner function whose derivative also appears.
- A complicated expression inside a square root or power often becomes u.
- In trig integrals, set u equal to the part whose derivative is nearby.
5. Examples
These examples show how substitution reduces the integral to a simpler form.
5.1. Example 1 — Polynomial Inside
Evaluate:
\int (3x^2 + 1)^5 (6x) \, dx
Let:
u = (3x^2 + 1) \Rightarrow du = 6x dx
Integral becomes:
\int u^5 \, du = \frac{u^6}{6} + C
Back-substituting gives:
\frac{(3x^2 + 1)^6}{6} + C
5.2. Example 2 — Trigonometric Substitution
Evaluate:
\int \cos x (\sin x)^4 dx
Let:
u = \sin x \Rightarrow du = \cos x \, dx
Integral reduces to:
\int u^4 du = \frac{u^5}{5} + C
Final answer:
\frac{(\sin x)^5}{5} + C
5.3. Example 3 — Exponential Substitution
Evaluate:
\int e^{3x} dx
Let:
u = 3x \Rightarrow du = 3 dx
So:
dx = \frac{du}{3}
The integral becomes:
\int e^u \frac{du}{3} = \frac{1}{3} e^u + C = \frac{e^{3x}}{3} + C
6. When Substitution Works Best
Substitution is especially useful when the integrand contains a composite function or when a piece of the integrand resembles the derivative of another part.