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The integral you provided is:

\[
\int \frac{x \sin(x^2)}{\cos(x^2)} \, dx
\]

This can be simplified using the following steps:

### Step 1: Simplify the integrand
The term \(\frac{\sin(x^2)}{\cos(x^2)}\) can be rewritten as \(\tan(x^2)\), so the integral becomes:

\[
\int x \tan(x^2) \, dx
\]

### Step 2: Substitution
Let \( u = x^2 \), which implies that \( du = 2x \, dx \) or \( \frac{du}{2} = x \, dx \).

Now substitute into the integral:

\[
\int \frac{1}{2} \tan(u) \, du
\]

### Step 3: Integrating \(\tan(u)\)
The integral of \(\tan(u)\) is:

\[
\int \tan(u) \, du = \ln|\sec(u)| + C
\]

Thus, the integral becomes:

\[
\frac{1}{2} \ln|\sec(u)| + C
\]

### Step 4: Substituting back \( u = x^2 \)
Now replace \( u \) with \( x^2 \):

\[
\frac{1}{2} \ln|\sec(x^2)| + C
\]

Thus, the solution to the integral is:

\[
\frac{1}{2} \ln|\sec(x^2)| + C
\]

Let me know if you want more details or have any questions!

### Related Questions:
1. How do you integrate functions involving trigonometric identities?
2. What is the significance of the substitution method in calculus?
3. How does the logarithmic function arise in trigonometric integrals?
4. Can this method be applied to other similar integrals involving \(\tan\) and \(\sec\)?
5. What other methods can be used for solving integrals involving trigonometric expressions?

### Tip:
Always look for opportunities to apply trigonometric identities to simplify integrals, especially when dealing with ratios of sine and cosine functions.

Solve the integral: ∫ (x * sin(x^2)) / cos(x^2) dx

Learn how to solve the integral ∫ (x * sin(x^2)) / cos(x^2) dx using substitution and trigonometric identities. This step-by-step guide covers key concepts from calculus, including substitution methods and integrating trigonometric functions. Suitable for university-level calculus students.

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