Calculus Examples

$f (x) = x + 1$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int x + 1 d x$

Step 3

Split the single integral into multiple integrals.

$\int x d x + \int d x$

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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

Step 7

The answer is the antiderivative of the function $f (x) = x + 1$ .

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