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.

This page provides a detailed solution to the integral ∫ 4x e^(8x^2 - 3) dx using the substitution method. Learn step-by-step how to solve this calculus problem, including the key substitution formula and how to integrate exponential functions. Suitable for students in Grades 11-12 or early college-level calculus.

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