To solve the integral:

$\int 4x e^{8x^2 - 3} \, dx$

We can use the substitution method.

Step 1: Substitution

Let: $u = 8x^2 - 3$

Now, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 16x$

This implies: $du = 16x \, dx$

We have $4x \, dx$ in the original integral, so we rewrite this as: $4x \, dx = \frac{du}{4}$

Step 2: Substituting into the integral

Now substitute $u$ and $du$ into the integral: $\int 4x e^{8x^2 - 3} \, dx = \int e^u \cdot \frac{du}{4}$

This simplifies to: $\frac{1}{4} \int e^u \, du$

Step 3: Integrate

The integral of $e^u$ is simply $e^u$: $\frac{1}{4} e^u + C$

Step 4: Substitute back for $u$

Finally, substitute $u = 8x^2 - 3$ back into the expression: $\frac{1}{4} e^{8x^2 - 3} + C$

Thus, the solution to the integral is: $\frac{1}{4} e^{8x^2 - 3} + C$

Would you like any further details or clarifications?

Related Questions:

1. How do you approach substitution in integrals involving exponential functions?
2. What other techniques could be useful for solving integrals like this?
3. How would you evaluate this integral over a definite range?
4. What happens if the exponent had a more complex form, e.g., $8x^3 - 3$?
5. Can integration by parts be used for this type of problem?

Tip: Substitution is especially helpful when the integrand contains a function and its derivative. Always check for this pattern.