Calculus Examples

$\frac{1}{2} x^{2} - 2 x + 3$

Step 1

Write $\frac{1}{2} x^{2} - 2 x + 3$ as a function.

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

Step 2

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

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

Step 3

Set up the integral to solve.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 7

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 8

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 9

Apply the constant rule.

$\frac{1}{2} (\frac{1}{3} x^{3} + C) - 2 (\frac{1}{2} x^{2} + C) + 3 x + C$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{2} (\frac{1}{3} x^{3} + C) - 2 (\frac{x^{2}}{2} + C) + 3 x + C$

Step 10.2

Simplify.

$\frac{x^{3}}{6} - x^{2} + 3 x + C$

Step 10.3

Reorder terms.

$\frac{1}{6} x^{3} - x^{2} + 3 x + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{1}{2} x^{2} - 2 x + 3$ .

$F (x) =$ $\frac{1}{6} x^{3} - x^{2} + 3 x + C$