Integration by Substitution

Integration by substitution, often called u-substitution, is a powerful technique in calculus used to evaluate integrals. You can think of it as the reverse of the chain rule for differentiation.

When you have an integral where the integrand contains a composite function and the derivative of the inner function is also present, substitution is the perfect tool.

The Method of U-Substitution

The general formula for substitution is based on the relationship: $\int f(g(x))g'(x) \, dx = \int f(u) \, du$

Here are the standard steps to solve an integral using substitution:

1. Choose an inner function $u = g(x)$. A good rule of thumb is to look for a part of the integrand whose derivative is also multiplying the rest of the expression.
2. Find the differential $du = g'(x) \, dx$.
3. Substitute $u$ and $du$ into the original integral to replace all instances of $x$ and $dx$.
4. Evaluate the new integral with respect to $u$.
5. Replace $u$ with the original expression $g(x)$ (for indefinite integrals).

Example 1: Indefinite Integral

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

1. Identify the inner function. Let $u = x^2$.
2. Find the derivative: $du = 2x \, dx$. Notice that $2x \, dx$ is exactly what we have in the original integral!
3. Substitute $u$ and $du$ into the integral: $\int \cos(u) \, du$
4. Evaluate the integral: $\int \cos(u) \, du = \sin(u) + C$
5. Substitute $x^2$ back in for $u$: $\sin(x^2) + C$

Example 2: Definite Integral

For definite integrals, you must also change the limits of integration to match your new variable $u$.

Evaluate $\int_0^1 x^2(x^3 + 1)^4 \, dx$

1. Let $u = x^3 + 1$.
2. Find the derivative: $du = 3x^2 \, dx$. Our integral only has $x^2 \, dx$, so we can rewrite this as $\frac{1}{3} \, du = x^2 \, dx$.
3. Change the limits of integration:
   • Lower limit: When $x = 0$, $u = (0)^3 + 1 = 1$.
   • Upper limit: When $x = 1$, $u = (1)^3 + 1 = 2$.
4. Substitute everything into the integral: $\int_1^2 (u)^4 \left(\frac{1}{3} \, du\right) = \frac{1}{3} \int_1^2 u^4 \, du$
5. Evaluate the definite integral: $\frac{1}{3} \left[ \frac{u^5}{5} \right]_1^2 = \frac{1}{15} (2^5 - 1^5) = \frac{1}{15} (32 - 1) = \frac{31}{15}$