Math Portfolio

Integration by Substitution

Understanding Integration by Substitution

Integration by substitution, also known as u-substitution, is a powerful method used to simplify complex integrals by introducing a new variable. This technique is essentially the reverse of the chain rule for differentiation.

The basic idea is to substitute $u = g(x)$ and $du = g'(x)dx$ to transform an integral of the form:

$$\int f(g(x)) \cdot g'(x) \, dx$$

into the simpler form:

$$\int f(u) \, du$$

This substitution works well when you can identify a composite function within the integral and its derivative (or a multiple of it) is also present.

Basic Steps for Integration by Substitution

  1. Identify a function $g(x)$ within the integrand that appears as part of a composite function.
  2. Set $u = g(x)$ and compute $du = g'(x) \, dx$.
  3. Check if $g'(x)$ appears in the integrand. If not, try to manipulate the integral to include it.
  4. Rewrite the integral in terms of $u$ and $du$.
  5. Evaluate the new integral in terms of $u$.
  6. Substitute back to express the result in terms of the original variable $x$.

Example

Evaluate $\int 2x \cos(x^2) \, dx$

Step 1: Let $u = x^2$, then $du = 2x \, dx$ or $dx = \frac{du}{2x}$

Step 2: Substitute into the integral:

$$\int 2x \cos(x^2) \, dx = \int 2x \cos(u) \cdot \frac{du}{2x} = \int \cos(u) \, du$$

Step 3: Evaluate the new integral:

$$\int \cos(u) \, du = \sin(u) + C$$

Step 4: Substitute back to get:

$$\int 2x \cos(x^2) \, dx = \sin(x^2) + C$$

Common Substitution Patterns

Trigonometric Functions

  • $\int \sin(ax) \, dx$ — Let $u = ax$
  • $\int \cos(ax) \, dx$ — Let $u = ax$
  • $\int \tan(ax) \, dx$ — Let $u = ax$
  • $\int x \sin(x^2) \, dx$ — Let $u = x^2$

Exponential Functions

  • $\int e^{ax} \, dx$ — Let $u = ax$
  • $\int x^n e^{ax^{n+1}} \, dx$ — Let $u = ax^{n+1}$
  • $\int \frac{1}{x} \, dx$ — Let $u = \ln(x)$
  • $\int \frac{f'(x)}{f(x)} \, dx$ — Let $u = f(x)$

Rational Functions

  • $\int \frac{1}{ax+b} \, dx$ — Let $u = ax+b$
  • $\int \frac{1}{x^2+a^2} \, dx$ — Let $u = \arctan(x/a)$
  • $\int \frac{1}{\sqrt{a^2-x^2}} \, dx$ — Let $u = \arcsin(x/a)$
  • $\int \frac{1}{x\sqrt{x^2-a^2}} \, dx$ — Let $u = \text{arcsec}(x/a)$

Notes and Resources

Key Takeaways

  • Integration by substitution is essentially the reverse of the chain rule for differentiation
  • Look for a function and its derivative (or a multiple of its derivative) in the integrand
  • Use pattern recognition to identify common substitution situations
  • Remember to transform both the function and the differential element $dx$
  • For definite integrals, either change the limits of integration or substitute back before evaluating