Calculus Examples

$f (x) = \frac{2 x^{3} - 3 x^{2} + 1}{x^{2}}$

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.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $2$ by $- 1$ .

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

Step 5

Expand $(2 x^{3} - 3 x^{2} + 1) x^{- 2}$ .

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

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

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

Step 5.4

Subtract $2$ from $3$ .

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

Step 5.5

Simplify.

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

Step 5.6

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

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

Step 5.7

Subtract $2$ from $2$ .

$\int 2 x - 3 x^{0} + 1 x^{- 2} d x$

Step 5.8

Anything raised to $0$ is $1$ .

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

Step 5.9

Multiply $- 3$ by $1$ .

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

Step 5.10

Multiply $x^{- 2}$ by $1$ .

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

Step 5.11

Move $- 3$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Apply the constant rule.

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

Step 12

Simplify.

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

Step 13

The answer is the antiderivative of the function $f (x) = \frac{2 x^{3} - 3 x^{2} + 1}{x^{2}}$ .

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