\int \:x^3\:\left(1\:+\:x^4\:\right)^3\:dx

To solve the integral

\[
\int x^3 (1 + x^4)^3 \, dx,
\]

we can use a substitution method. Let’s proceed step by step.

### Step 1: Substitution
Let \( u = 1 + x^4 \). Then, differentiate both sides with respect to \( x \):

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

### Step 2: Rewrite the integral
Substitute \( u = 1 + x^4 \) and \( x^3 \, dx = \frac{du}{4} \) into the original integral:

\[
\int x^3 (1 + x^4)^3 \, dx = \int (u^3) \cdot \frac{du}{4}.
\]

This simplifies to:

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

### Step 3: Integrate
Now integrate \( u^3 \):

\[
\frac{1}{4} \int u^3 \, du = \frac{1}{4} \cdot \frac{u^4}{4} = \frac{u^4}{16}.
\]

### Step 4: Substitute back \( u = 1 + x^4 \)
Finally, replace \( u \) with \( 1 + x^4 \):

\[
\frac{(1 + x^4)^4}{16} + C.
\]

### Final Answer:
\[
\int x^3 (1 + x^4)^3 \, dx = \frac{(1 + x^4)^4}{16} + C.
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you solve \(\int x^5 (1 + x^6)^2 \, dx\)?
2. Can you solve \(\int x^2 (1 + x^3)^5 \, dx\) using the same substitution method?
3. What happens if you use substitution in \(\int (1 + x^2)^n \, dx\)?
4. How do you differentiate \((1 + x^4)^4\) and verify the answer above?
5. Can you apply integration by parts to this problem instead?

**Tip:** When dealing with powers of binomials inside an integral, substitution is often the most efficient approach.

Learn how to solve the integral \int x^3 (1 + x^4)^3 dx step-by-step using the substitution method. This calculus problem involves key concepts in integration and is suitable for Grades 11-12 and college-level calculus.

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