To solve the integral $\int 2x e^{5x^2} \, dx$, we can use the substitution method.

Step 1: Choose a substitution

Let’s choose $u = 5x^2$. This simplifies the exponent inside the exponential function.

Now, differentiate $u$ with respect to $x$:

$\frac{du}{dx} = 10x \quad \Rightarrow \quad du = 10x \, dx$

Thus, we can express $2x \, dx$ as:

$2x \, dx = \frac{1}{5} du$

Step 2: Substitute into the integral

Now substitute into the original integral:

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

Step 3: Solve the integral

The integral of $e^u$ is simply $e^u$, so we get:

$\frac{1}{5} \int e^u \, du = \frac{1}{5} e^u + C$

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

Now, replace $u$ with $5x^2$ to get the final answer:

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

Final Answer:

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

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Do you need further clarification on any of the steps, or is there a specific concept you'd like me to elaborate on?

Here are 5 related questions:

1. How does the method of substitution work for integrals?
2. What is the general form for the integration of an exponential function?
3. Can we solve the same problem using integration by parts instead of substitution?
4. How do we handle integrals involving higher powers of $x$ in the exponent?
5. What is the difference between indefinite and definite integrals in the context of this problem?

Tip: When working with substitution, always check if the derivative of one part of the integrand matches another part to make the substitution smoother.