Calculus Examples

$\frac{4}{x^{2}} + 2 x - 3 x^{2}$

Step 1

Write $\frac{4}{x^{2}} + 2 x - 3 x^{2}$ as a function.

$f (x) = \frac{4}{x^{2}} + 2 x - 3 x^{2}$

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.

$F (x) = \int \frac{4}{x^{2}} + 2 x - 3 x^{2} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{4}{x^{2}} d x + \int 2 x d x + \int - 3 x^{2} d x$

Step 5

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

$4 \int \frac{1}{x^{2}} d x + \int 2 x d x + \int - 3 x^{2} d x$

Step 6

Apply basic rules of exponents.

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Step 6.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 \int x^{2 \cdot - 1} d x + \int 2 x d x + \int - 3 x^{2} d x$

Step 6.2.2

Multiply $2$ by $- 1$ .

$4 \int x^{- 2} d x + \int 2 x d x + \int - 3 x^{2} d x$

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Simplify.

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Step 12.2.1

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

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

Step 12.2.2

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

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

Step 12.2.3

Cancel the common factor of $- 3$ and $3$ .

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Step 12.2.3.1

Factor $3$ out of $- 3 x^{3}$ .

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

Step 12.2.3.2

Cancel the common factors.

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Step 12.2.3.2.1

Factor $3$ out of $3$ .

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

Step 12.2.3.2.2

Cancel the common factor.

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

Step 12.2.3.2.3

Rewrite the expression.

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

Step 12.2.3.2.4

Divide $- x^{3}$ by $1$ .

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

Step 13

The answer is the antiderivative of the function $f (x) = \frac{4}{x^{2}} + 2 x - 3 x^{2}$ .

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