Calculus Examples

$x^{3} + x + 102$

Step 1

Write $x^{3} + x + 102$ as a function.

$f (x) = x^{3} + x + 102$

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.

$F (x) = \int x^{3} + x + 102 d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{3} d x + \int x d x + \int 102 d x$

Step 5

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

$\frac{1}{4} x^{4} + C + \int x d x + \int 102 d x$

Step 6

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

$\frac{1}{4} x^{4} + C + \frac{1}{2} x^{2} + C + \int 102 d x$

Step 7

Apply the constant rule.

$\frac{1}{4} x^{4} + C + \frac{1}{2} x^{2} + C + 102 x + C$

Step 8

Simplify.

$\frac{1}{4} x^{4} + \frac{1}{2} x^{2} + 102 x + C$

Step 9

The answer is the antiderivative of the function $f (x) = x^{3} + x + 102$ .

$F (x) =$ $\frac{1}{4} x^{4} + \frac{1}{2} x^{2} + 102 x + C$