Methods of Integration

Basic Steps for $u$-Substitution

#What is integration by substitution?

Substitution simplifies an integral by defining an alternative variable (usually $u$) in terms of the original variable (usually $x$). The new integral in $u$ is often easier to solve, and the substitution can be reversed at the end to express the answer in terms of $x$.

#How do I integrate simple functions using $u$-substitution?

In a simple integral involving substitution, you will usually be integrating a composite function (i.e., 'function of a function'). Substitution can be a safer method when integrating by inspection is awkward or difficult to spot.

Steps for $u$-Substitution:

1. Identify the substitution to be used – it will be the secondary (or 'inside') function in a composite function.

   • If the integral involves $f(g(x))$, let $u = g(x)$.
   • Example: $\int 3x \cos(x^2 - 5) , dx$
     • Let $u = x^2 - 5$.

2. Differentiate the substitution and rearrange.

   • $\frac{du}{dx}$ can be treated like a fraction. Multiply by $dx$ to get rid of fractions.
   • Example: $u = x^2 - 5 \Rightarrow \frac{du}{dx} = 2x$
     • Then $du = 2x , dx \Rightarrow x , dx = \frac{1}{2} , du$.

3. Replace all parts of the integral.

   • Replace all $u$ terms, including $dx$.
   • Example: $\int 3x \cos(x^2 - 5) , dx = 3 \int \cos(x^2 - 5) \cdot x , dx$
     • So $\int 3x \cos(x^2 - 5) , dx = 3 \int \cos(u) \cdot \frac{1}{2} , du = \frac{3}{2} \int \cos(u) , du$.

4. Integrate.

   • Example: ( \frac{3}{2} \int \cos(u) , du = ...