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.