Find the Function f(x)=x^2+5x

$f (x) = x^{2} + 5 x$

Step 1

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

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

Step 2

Split the single integral into multiple integrals.

$\int x^{2} d x + \int 5 x d x$

Step 3

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

$\frac{1}{3} x^{3} + C + \int 5 x d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{1}{3} x^{3} + C + 5 \int x d x$

Step 5

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

$\frac{1}{3} x^{3} + C + 5 (\frac{1}{2} x^{2} + C)$

Step 6

Simplify.

$\frac{1}{3} x^{3} + 5 (\frac{1}{2}) x^{2} + C$

Combine $5$ and $\frac{1}{2}$ .

$\frac{1}{3} x^{3} + \frac{5}{2} x^{2} + C$

Step 7

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

$F (x) = \frac{1}{3} x^{3} + \frac{5}{2} x^{2} + C$