The given integral is:

$\int x e^{5x^2} \, dx$

Solution:

To solve this, we use substitution.

1. Let $u = 5x^2$. Then, $du = 10x \, dx$, which implies $x \, dx = \frac{1}{10} \, du$.

2. Substitute into the integral:

$\int x e^{5x^2} \, dx = \int e^u \cdot \frac{1}{10} \, du = \frac{1}{10} \int e^u \, du$

3. The integral of $e^u$ is $e^u$. So:

$\frac{1}{10} \int e^u \, du = \frac{1}{10} e^u + C$

4. Substitute back $u = 5x^2$:

$\frac{1}{10} e^u + C = \frac{1}{10} e^{5x^2} + C$

Final Answer:

$\int x e^{5x^2} \, dx = \frac{1}{10} e^{5x^2} + C$

Would you like further explanations or a step-by-step breakdown of substitution?

Related Questions:

1. What is the method of substitution, and when is it used?
2. How do you solve integrals involving exponential functions like $e^{ax^2}$?
3. Can integration by parts be applied to this problem? Why or why not?
4. How do definite integrals differ from indefinite ones in exponential cases?
5. What is the role of constants like $C$ in indefinite integration?

Tip: Always check if substitution simplifies the integrand before attempting other methods!