Math Snap

STEP 1

1. We are given the function $f(x) = \int e^{x^3} \, dx$.
2. We need to find the Taylor polynomial with 4 nonzero terms about $x = 0$.
3. The Taylor polynomial is centered at $x = 0$, which is also known as the Maclaurin series.
4. We are given that $f(0) = 0$.

STEP 2

1. Find the Taylor series expansion of $e^{x^3}$ about $x = 0$.
2. Integrate the series term-by-term to find the Taylor series for $f(x)$.
3. Identify the first four nonzero terms of the Taylor series for $f(x)$.
4. Write the Taylor polynomial using these terms.

STEP 3

Start with the Taylor series expansion for $e^u$ at $u = 0$:
$$e^u = 1 + u + \frac{u^2}{2!} + \frac{u^3}{3!} + \cdots$$ Substitute $u = x^3$ to find the series for $e^{x^3}$:
$$e^{x^3} = 1 + x^3 + \frac{(x^3)^2}{2!} + \frac{(x^3)^3}{3!} + \cdots$$ $$e^{x^3} = 1 + x^3 + \frac{x^6}{2} + \frac{x^9}{6} + \cdots$$

STEP 4

Integrate the series term-by-term to find $f(x) = \int e^{x^3} \, dx$:
$$f(x) = \int \left( 1 + x^3 + \frac{x^6}{2} + \frac{x^9}{6} + \cdots \right) \, dx$$ $$f(x) = x + \frac{x^4}{4} + \frac{x^7}{14} + \frac{x^{10}}{60} + \cdots$$

STEP 5

Identify the first four nonzero terms of the Taylor series for $f(x)$:
1. $x$
2. $\frac{x^4}{4}$
3. $\frac{x^7}{14}$
4. $\frac{x^{10}}{60}$

Write the Taylor polynomial using these terms:
The Taylor polynomial with 4 nonzero terms is:
$$P(x) = x + \frac{x^4}{4} + \frac{x^7}{14} + \frac{x^{10}}{60}$$ The Taylor polynomial with 4 nonzero terms for $f(x)$ about $x = 0$ is:
$$\boxed{x + \frac{x^4}{4} + \frac{x^7}{14} + \frac{x^{10}}{60}}$$