Add x= 1 to x = 3 I want to know to change back when I have to change from u to x

To evaluate a definite integral by substitution (also known as **u-substitution**), it is crucial to remember to **change the limits of integration** from the \(x\)-values to the corresponding \(u\)-values. If the problem provides the limits of integration in terms of \(x\) (e.g., from \(x = 1\) to \(x = 3\)), we must find the corresponding limits in terms of \(u\) after substituting \(u = g(x)\). After computing the integral in terms of \(u\), you can either convert back to \(x\) or evaluate using the new limits.

Let's go through an example step-by-step to clarify when and how to change back from \(u\) to \(x\):

### Example

Suppose we want to evaluate the following integral:

\[
\int_{1}^{3} 2x \sqrt{x^2 + 1} \, dx
\]

**Step 1: Choose a Substitution**

We choose a substitution that simplifies the integrand. In this case, let's set:

\[
u = x^2 + 1
\]

**Step 2: Compute \(du\) and Substitute**

Differentiate \(u\) with respect to \(x\):

\[
\frac{du}{dx} = 2x \implies du = 2x \, dx
\]

Notice that \(2x \, dx = du\), which matches a part of our integrand. Thus, the integral becomes:

\[
\int_{1}^{3} \sqrt{x^2 + 1} \cdot 2x \, dx = \int \sqrt{u} \, du
\]

**Step 3: Change the Limits of Integration**

Now, we need to change the limits from \(x\)-values to \(u\)-values. When \(x = 1\):

\[
u = 1^2 + 1 = 2
\]

And when \(x = 3\):

\[
u = 3^2 + 1 = 10
\]

So, the new limits for the integral in terms of \(u\) are from \(u = 2\) to \(u = 10\). The integral now becomes:

\[
\int_{2}^{10} \sqrt{u} \, du
\]

**Step 4: Integrate with Respect to \(u\)**

Compute the integral in terms of \(u\):

\[
\int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C
\]

**Step 5: Apply the New Limits and Evaluate**

Now, apply the limits \(u = 2\) to \(u = 10\):

\[
\left[ \frac{2}{3} u^{3/2} \right]_{2}^{10} = \frac{2}{3} (10^{3/2}) - \frac{2}{3} (2^{3/2})
\]

Compute the values:

\[
= \frac{2}{3} (10 \sqrt{10}) - \frac{2}{3} (2 \sqrt{2})
\]

\[
= \frac{20 \sqrt{10}}{3} - \frac{4 \sqrt{2}}{3}
\]

This is the final evaluated integral.

**Note**: **You do not need to change back to \(x\) if you have already adjusted the limits to match your new \(u\)-variable**. The integral is fully evaluated in terms of \(u\) with its corresponding limits.

### Conclusion

- **Change the limits of integration** when substituting: if your integral is in terms of \(u\), your limits must be in terms of \(u\) as well.
- **If you do not change the limits**, you must substitute back to \(x\) before evaluating the integral.

Do you have any specific questions about this process, or would you like to see more examples?

### 5 Related Questions to Expand Your Understanding:

1. What are some common substitutions to simplify integrals?
2. How does trigonometric substitution work for integrals involving \(\sqrt{a^2 - x^2}\)?
3. When is it beneficial to use integration by parts instead of substitution?
4. How can you handle improper integrals with infinite limits or discontinuities?
5. What is the difference between definite and indefinite integrals in terms of evaluation and substitution?

### Tip:
Always write down the new limits when substituting variables in a definite integral. This prevents errors and helps streamline your integration process.

Learn how to evaluate a definite integral using substitution (u-substitution) method with step-by-step guidance. Understand when and how to change limits of integration from x-values to u-values. Example provided for clarity.

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