Find the Function f(x)=(x-1)^3+2

$f (x) = {(x - 1)}^{3} + 2$

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Split the single integral into multiple integrals.

$\int {(x - 1)}^{3} d x + \int 2 d x$

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Let $u = x - 1$ . Find $\frac{d u}{d x}$ .

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 1]$

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Add $1$ and $0$ .

$1$

Rewrite the problem using $u$ and $d u$ .

$\int u^{3} d u + \int 2 d x$

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{4} u^{4} + C + \int 2 d x$

Apply the constant rule.

$\frac{1}{4} u^{4} + C + 2 x + C$

Simplify.

$\frac{1}{4} u^{4} + 2 x + C$

Replace all occurrences of $u$ with $x - 1$ .

$\frac{1}{4} {(x - 1)}^{4} + 2 x + C$

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$F (x) = \frac{1}{4} \cdot {(x - 1)}^{4} + 2 x + C$