Answer
See explanation
Work Step by Step
The method of substitution for definite integrals involves making a change of variables to simplify the integral and then adjusting the limits of integration accordingly. The substitution allows you to replace the original variable with a new one, making the integration more manageable.
Here's a general outline of the steps for substitution in definite integrals:
1. Choose a substitution:Pick a suitable substitution to simplify the integral. Common choices involve trigonometric, algebraic, or exponential functions.
2. Calculate differentials: Find the differential of the substitution variable, typically denoted as \(du\).
3. Change limits of integration: Replace the original limits of integration with the corresponding values in terms of the new variable.
4. Perform the substitution: Substitute the new variable and its differential into the integrand.
5. Evaluate the integral:Integrate the expression with respect to the new variable.
6. Convert back: Express the final result in terms of the original variable.
Let's illustrate this with an example:
Example:
Certainly! Let's consider another example:
\[ \int_{0}^{1} \frac{2x}{(1 + x^2)^2} \, dx \]
Substitution:
Let \( u = 1 + x^2 \), then \( du = 2x \, dx \).
Adjust limits:
When \( x = 0 \), \( u = 1 + 0^2 = 1 \), and when \( x = 1 \), \( u = 1 + 1^2 = 2 \).
Substitute:
\[ \int_{0}^{1} \frac{2x}{(1 + x^2)^2} \, dx = \frac{1}{2} \int_{1}^{2} \frac{1}{u^2} \, du \]
Evaluate:
\[ \frac{1}{2} \int_{1}^{2} \frac{1}{u^2} \, du = \frac{1}{2} \left[ -\frac{1}{u} \right]_{1}^{2} = \frac{1}{2} \left(-\frac{1}{2} + 1\right) \]
So, \[ \int_{0}^{1} \frac{2x}{(1 + x^2)^2} \, dx = \frac{1}{4} \].
