Answer
See the explanation
Work Step by Step
Evaluating indefinite integrals by substitution, also known as the u-substitution method, involves substituting a new variable \( u \) for a portion of the integrand. This substitution simplifies the integral and makes it easier to evaluate. The general process involves three steps:
1. Choose an appropriate substitution: Identify a portion of the integrand that resembles a function and its derivative, so that substitution will simplify the integral.
2. Compute \( du \): Find the differential of the new variable \( u \) with respect to the original variable \( x \).
3. Perform the substitution: Replace all occurrences of the original variable \( x \) and its differentials with the new variable \( u \) and its differential, respectively. Then, integrate the new expression with respect to \( u \), and finally, substitute back in terms of \( x \) if necessary.
Here's an example to illustrate the process:
Example:
Evaluate the indefinite integral \( \int 2x \cos(x^2) dx \).
Solution:
1. Choose an appropriate substitution: Let \( u = x^2 \), so that \( du = 2x dx \).
2. Compute \( du \): We already have \( du = 2x dx \).
3. Perform the substitution:
Substituting \( u = x^2 \) and \( du = 2x dx \) into the integral, we get:
\[ \int 2x \cos(x^2) dx = \int \cos(u) du \]
This simplifies the integral significantly.
Now, integrating \( \cos(u) \) with respect to \( u \), we get \( \sin(u) + C \), where \( C \) is the constant of integration.
Finally, substituting back \( u = x^2 \), we get:
\[ \int 2x \cos(x^2) dx = \sin(x^2) + C \]
So, \( \sin(x^2) + C \) is the antiderivative of \( 2x \cos(x^2) \) with respect to \( x \).
This example demonstrates how substitution can simplify the integral and make it more manageable for evaluation.
