https://file.aiquickdraw.com/mathbot/file/1725384731995c5sow4ds.jpg

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!

Evaluate the integral from 1 to 2 of x*sin(x^2) dx.

Solve the definite integral of x*sin(x^2) from 1 to 2 using the substitution method. This problem involves key calculus concepts like integration and the substitution method, ideal for students in Calculus I.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation