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The given image appears to show an integral problem along with multiple-choice answers. Let me transcribe the integral and the provided options:

**Integral:**  
\[
\int x^3 e^{x^4} \, dx
\]

**Options:**
1. \(\frac{e^{x^4}}{4} + C\)
2. \(\frac{e^{x^4}}{3} + C\)
3. \(\frac{e^{x^4}}{5} + C\)
4. \(\frac{e^{x^4}}{6} + C\)

This integral looks like it requires a **substitution** method. Let's solve it step by step.

### Step 1: Substitution
Let’s choose \( u = x^4 \). Then, the derivative of \( u \) is:
\[
du = 4x^3 \, dx \quad \Rightarrow \quad \frac{du}{4} = x^3 \, dx
\]

### Step 2: Rewrite the integral
Substitute \( u = x^4 \) and \( x^3 \, dx = \frac{du}{4} \):
\[
\int x^3 e^{x^4} \, dx = \int e^u \cdot \frac{du}{4} = \frac{1}{4} \int e^u \, du
\]

### Step 3: Integrate
The integral of \( e^u \) is simply \( e^u \):
\[
\frac{1}{4} \int e^u \, du = \frac{1}{4} e^u + C
\]

### Step 4: Substitute back \( u = x^4 \)
\[
\frac{1}{4} e^{x^4} + C
\]

### Conclusion
Thus, the solution to the integral is:
\[
\frac{e^{x^4}}{4} + C
\]

Therefore, the correct answer is **Option 1: \(\frac{e^{x^4}}{4} + C\)**.

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Do you want more details or have any other questions?

Here are 5 related questions:
1. How does substitution work in integration?
2. Can substitution always simplify exponentials in integrals?
3. What if the power of \( x \) in the integral was different?
4. What is the difference between indefinite and definite integrals?
5. How would the integral change if the function were \( e^{x^3} \)?

**Tip:** When dealing with integrals involving \( e^x \), always check if substitution can simplify the exponential term!

Learn how to solve the integral ∫ x^3 e^(x^4) dx using the substitution method. This detailed solution covers the key steps, including selecting u = x^4 and integrating e^u, suitable for students in Calculus I.

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