# Substitution Method

### Substitution Method for Indefinite Integrals:

• Principle: Involves substituting variables to simplify complex integrals.
• Steps:
1. Choose Substitution: Select a suitable substitution involving a new variable, often denoted by u.
2. Compute Differential: Find du = f'(x) dx, where f'(x) represents the derivative of the chosen substitution variable u with respect to x.
3. Substitute: Express the integral in terms of the new variable u and du.
4. Integrate: Solve the integral in terms of u.
5. Resubstitute: Convert the solution back to the original variable x.

### Example:

Let's solve the indefinite integral ∫ 2x${\left({x}^{2}+1\right)}^{5}$ dx using the substitution method:

### Steps:

1. Choose Substitution: Let $u={x}^{2}+1$.

2. Compute Differential: Find $du=2xdx$ by differentiating $u$ with respect to $x$.

3. Substitute: Rewrite the integral in terms of $u$ and $du$:

$\int 2x\left({x}^{2}+1{\right)}^{5}dx=\int {u}^{5}du$

4. Integrate: Now, integrate the expression in terms of $u$:

$\int {u}^{5}du=\frac{{u}^{6}}{6}+C$

5. Resubstitute: Replace $u$ with the original substitution:

$\frac{\left({x}^{2}+1{\right)}^{6}}{6}+C$

### Tips for Substitution Method:

• Appropriate Choices: Select substitutions that simplify the integral or reduce it to a standard form.
• Derivative Alignment: Ensure the presence of the differential $du$ in the integral to match with the derivative of the substituted variable.
• Back-substitution: Always express the final result in terms of the original variable.