Calculus Examples

$f (x) = 4 x - 6 x^{2} + 1$

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.

$F (x) = \int 4 x - 6 x^{2} + 1 d x$

Step 3

Split the single integral into multiple integrals.

$\int 4 x d x + \int - 6 x^{2} d x + \int d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int x d x + \int - 6 x^{2} d x + \int d x$

Step 5

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

$4 (\frac{1}{2} x^{2} + C) + \int - 6 x^{2} d x + \int d x$

Step 6

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$4 (\frac{1}{2} x^{2} + C) - 6 \int x^{2} d x + \int d x$

Step 7

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

$4 (\frac{1}{2} x^{2} + C) - 6 (\frac{1}{3} x^{3} + C) + \int d x$

Step 8

Apply the constant rule.

$4 (\frac{1}{2} x^{2} + C) - 6 (\frac{1}{3} x^{3} + C) + x + C$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

$4 (\frac{x^{2}}{2} + C) - 6 (\frac{1}{3} x^{3} + C) + x + C$

Step 9.1.2

Combine $\frac{1}{3}$ and $x^{3}$ .

$4 (\frac{x^{2}}{2} + C) - 6 (\frac{x^{3}}{3} + C) + x + C$

Step 9.2

Simplify.

$2 x^{2} - 2 x^{3} + x + C$

Step 10

The answer is the antiderivative of the function $f (x) = 4 x - 6 x^{2} + 1$ .

$F (x) =$ $2 x^{2} - 2 x^{3} + x + C$