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We are given the integral:

\[
\int e^{x^4} \, dx
\]

and instructed to apply the substitution \( u = x^4 \).

### Step 1: Differentiate the substitution
If \( u = x^4 \), then:

\[
\frac{du}{dx} = 4x^3 \quad \text{or} \quad du = 4x^3 \, dx
\]

Thus, solving for \( dx \), we get:

\[
dx = \frac{du}{4x^3}
\]

### Step 2: Substitute into the integral
Now, we need to express everything in terms of \( u \) and \( du \). The integral \( \int e^{x^4} \, dx \) becomes:

\[
\int e^u \cdot \frac{du}{4x^3}
\]

But we still have \( x^3 \) to deal with. From \( u = x^4 \), we know that:

\[
x = u^{1/4}
\]

Thus:

\[
x^3 = u^{3/4}
\]

Substitute this into the integral:

\[
\int e^u \cdot \frac{du}{4u^{3/4}}
\]

### Step 3: Simplify the integral
We can factor out \( \frac{1}{4} \), so the integral becomes:

\[
\frac{1}{4} \int \frac{e^u}{u^{3/4}} \, du
\]

This is the resulting integral after the substitution \( u = x^4 \).

Would you like to go into more detail on this result, or explore further? Let me know if you have any questions!

### 5 Related Questions:
1. How do you solve the integral \( \int \frac{e^u}{u^{3/4}} \, du \)?
2. What techniques can be used to solve integrals involving exponentials and powers of \( u \)?
3. How does substitution help simplify complex integrals?
4. What are some common substitutions for integrals involving \( x^n \)?
5. Can you explain how to perform integration by parts on integrals involving exponential functions?

### Tip:
When using substitution in integrals, always ensure that all terms (including \( dx \)) are properly rewritten in terms of the new variable. This step ensures a smooth transition to the new integral form.

Consider the integral ∫ e^(x^4) dx. Apply the substitution u = x^4. What is the resulting integral?

Learn how to solve the integral ∫ e^(x^4) dx using the substitution method, u = x^4. Follow step-by-step instructions to transform and simplify the integral, suitable for students studying undergraduate calculus.

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