The integral given is:

$\int_{1}^{2} x \sin(x^2) \, dx$

This is an example of an integral that can be solved using the method of substitution. Let's solve it step by step.

Step 1: Substitution

Let $u = x^2$. Then, $du = 2x \, dx$ or $\frac{du}{2} = x \, dx$.

Step 2: Change of Limits

Since $u = x^2$, the limits of integration change as follows:

• When $x = 1$, $u = 1^2 = 1$.
• When $x = 2$, $u = 2^2 = 4$.

Step 3: Substitute into the Integral

Substituting $u$ and $du$ into the integral:

$\int_{1}^{2} x \sin(x^2) \, dx = \int_{1}^{4} \sin(u) \frac{du}{2}$

This simplifies to:

$\frac{1}{2} \int_{1}^{4} \sin(u) \, du$

Step 4: Integrate

The integral of $\sin(u)$ is $-\cos(u)$. So:

$\frac{1}{2} \left[ -\cos(u) \right]_{1}^{4}$

Step 5: Apply the Limits

Now, apply the limits:

$\frac{1}{2} \left[ -\cos(4) + \cos(1) \right]$

Therefore, the final answer is:

$\frac{1}{2} \left[ \cos(1) - \cos(4) \right]$

Would you like more details on any of the steps or have any other questions?

Here are five related questions you might find interesting:

1. How does substitution work for integrals with more complex functions?
2. What are some common functions where substitution is especially useful?
3. How can we evaluate integrals when substitution is not straightforward?
4. What are definite integrals, and how do they differ from indefinite integrals?
5. How would you solve an integral with limits that are not straightforward, like in polar coordinates?

Tip: When using substitution for definite integrals, always remember to change the limits of integration according to the new variable!