Explain integration by substitution with an example.
Understand the Problem
The question is asking about the integration by substitution method. This suggests a need for an explanation of the technique itself, perhaps with a simple example to illustrate its application.
Answer
$\sin(x^2) + C$
Answer for screen readers
The integral of $2x \cdot \cos(x^2)$ is $\sin(x^2) + C$.
Steps to Solve
- Introduce Integration by Substitution
Integration by substitution, also known as u-substitution, is a technique used to simplify integrals by replacing a part of the integrand with a new variable, $u$. This often transforms a complex integral into a simpler one that can be easily evaluated.
- Explain the substitution process
The core idea is to find a function $g(x)$ within the integrand whose derivative $g'(x)$ is also present (up to a constant multiple). We then let $u = g(x)$, so $du = g'(x) dx$. Substituting these into the integral can simplify the expression.
- Illustrative example
Let's consider the integral: $$ \int 2x \cdot \cos(x^2) , dx $$
- Identify $g(x)$ and $g'(x)$
Notice that the derivative of $x^2$ is $2x$, which is also present in the integral. So, let's try: $u = x^2$, then $\frac{du}{dx} = 2x$, which means $du = 2x , dx$.
- Perform the substitution
Substitute $u$ and $du$ into the original integral: $$ \int \cos(u) , du $$
- Evaluate the simplified integral
The integral of $\cos(u)$ is $\sin(u)$: $$ \int \cos(u) , du = \sin(u) + C $$ where $C$ is the constant of integration.
- Substitute back to the original variable
Finally, substitute $x^2$ back in for $u$: $$ \sin(x^2) + C $$
- Final Result
Therefore, the integral of $2x \cdot \cos(x^2)$ is $\sin(x^2) + C$.
The integral of $2x \cdot \cos(x^2)$ is $\sin(x^2) + C$.
More Information
Integration by substitution is the reverse of the chain rule in differentiation. It's a powerful technique for simplifying integrals that would otherwise be difficult to solve directly. The key is to identify a suitable substitution that simplifies the integrand.
Tips
- Forgetting to substitute back to the original variable after integrating with respect to $u$.
- Incorrectly identifying the substitution $u$ and its derivative $du$.
- Forgetting the constant of integration, $C$.
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