Calculus Examples

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

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.

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

Step 3

Simplify.

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Step 3.1

Factor $x^{2}$ out of $3 x^{4} - x^{3} + 6 x^{2}$ .

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Step 3.1.1

Factor $x^{2}$ out of $3 x^{4}$ .

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

Step 3.1.2

Factor $x^{2}$ out of $- x^{3}$ .

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

Step 3.1.3

Factor $x^{2}$ out of $6 x^{2}$ .

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

Step 3.1.4

Factor $x^{2}$ out of $x^{2} (3 x^{2}) + x^{2} (- x)$ .

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

Step 3.1.5

Factor $x^{2}$ out of $x^{2} (3 x^{2} - x) + x^{2} \cdot 6$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\int \frac{3 x^{2} - x + 6}{x^{2}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

$\int (3 x^{2} - x + 6) x^{- 2} d x$

Step 5

Expand $(3 x^{2} - x + 6) x^{- 2}$ .

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Step 5.1

Apply the distributive property.

$\int (3 x^{2} - x) x^{- 2} + 6 x^{- 2} d x$

Step 5.2

Apply the distributive property.

$\int 3 x^{2} x^{- 2} - x \cdot x^{- 2} + 6 x^{- 2} d x$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 3 x^{2 - 2} - x \cdot x^{- 2} + 6 x^{- 2} d x$

Step 5.4

Subtract $2$ from $2$ .

$\int 3 x^{0} - x \cdot x^{- 2} + 6 x^{- 2} d x$

Step 5.5

Anything raised to $0$ is $1$ .

$\int 3 \cdot 1 - x \cdot x^{- 2} + 6 x^{- 2} d x$

Step 5.6

Multiply $3$ by $1$ .

$\int 3 - x \cdot x^{- 2} + 6 x^{- 2} d x$

Step 5.7

Factor out negative.

$\int 3 - (x \cdot x^{- 2}) + 6 x^{- 2} d x$

Step 5.8

Raise $x$ to the power of $1$ .

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

Step 5.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 3 - x^{1 - 2} + 6 x^{- 2} d x$

Step 5.10

Subtract $2$ from $1$ .

$\int 3 - x^{- 1} + 6 x^{- 2} d x$

Step 5.11

Reorder $3$ and $- x^{- 1}$ .

$\int - x^{- 1} + 3 + 6 x^{- 2} d x$

Step 5.12

Move $3$ .

$\int - x^{- 1} + 6 x^{- 2} + 3 d x$

Step 6

Split the single integral into multiple integrals.

$\int - x^{- 1} d x + \int 6 x^{- 2} d x + \int 3 d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int x^{- 1} d x + \int 6 x^{- 2} d x + \int 3 d x$

Step 8

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

$- (\ln (| x |) + C) + \int 6 x^{- 2} d x + \int 3 d x$

Step 9

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

$- (\ln (| x |) + C) + 6 \int x^{- 2} d x + \int 3 d x$

Step 10

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

$- (\ln (| x |) + C) + 6 (- x^{- 1} + C) + \int 3 d x$

Step 11

Apply the constant rule.

$- (\ln (| x |) + C) + 6 (- x^{- 1} + C) + 3 x + C$

Step 12

Simplify.

$- \ln (| x |) - \frac{6}{x} + 3 x + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{3 x^{4} - x^{3} + 6 x^{2}}{x^{4}}$ .

$F (x) =$ $- \ln (| x |) - \frac{6}{x} + 3 x + C$