Calculus Examples

$y = \frac{3 x^{3} - 7 x^{2} + 4}{x^{2}}$

Step 1

Write $y = \frac{3 x^{3} - 7 x^{2} + 4}{x^{2}}$ as a function.

$f (x) = \frac{3 x^{3} - 7 x^{2} + 4}{x^{2}}$

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.

$F (x) = \int \frac{3 x^{3} - 7 x^{2} + 4}{x^{2}} d x$

Step 4

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

$\int (3 x^{3} - 7 x^{2} + 4) {(x^{2})}^{- 1} d x$

Step 5

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

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Step 5.1

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

$\int (3 x^{3} - 7 x^{2} + 4) x^{2 \cdot - 1} d x$

Step 5.2

Multiply $2$ by $- 1$ .

$\int (3 x^{3} - 7 x^{2} + 4) x^{- 2} d x$

Step 6

Expand $(3 x^{3} - 7 x^{2} + 4) x^{- 2}$ .

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Step 6.1

Apply the distributive property.

$\int (3 x^{3} - 7 x^{2}) x^{- 2} + 4 x^{- 2} d x$

Step 6.2

Apply the distributive property.

$\int 3 x^{3} x^{- 2} - 7 x^{2} x^{- 2} + 4 x^{- 2} d x$

Step 6.3

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

$\int 3 x^{3 - 2} - 7 x^{2} x^{- 2} + 4 x^{- 2} d x$

Step 6.4

Subtract $2$ from $3$ .

$\int 3 x^{1} - 7 x^{2} x^{- 2} + 4 x^{- 2} d x$

Step 6.5

Simplify.

$\int 3 x - 7 x^{2} x^{- 2} + 4 x^{- 2} d x$

Step 6.6

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

$\int 3 x - 7 x^{2 - 2} + 4 x^{- 2} d x$

Step 6.7

Subtract $2$ from $2$ .

$\int 3 x - 7 x^{0} + 4 x^{- 2} d x$

Step 6.8

Anything raised to $0$ is $1$ .

$\int 3 x - 7 \cdot 1 + 4 x^{- 2} d x$

Step 6.9

Multiply $- 7$ by $1$ .

$\int 3 x - 7 + 4 x^{- 2} d x$

Step 6.10

Move $- 7$ .

$\int 3 x + 4 x^{- 2} - 7 d x$

Step 7

Split the single integral into multiple integrals.

$\int 3 x d x + \int 4 x^{- 2} d x + \int - 7 d x$

Step 8

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

$3 \int x d x + \int 4 x^{- 2} d x + \int - 7 d x$

Step 9

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

$3 (\frac{1}{2} x^{2} + C) + \int 4 x^{- 2} d x + \int - 7 d x$

Step 10

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

$3 (\frac{1}{2} x^{2} + C) + 4 \int x^{- 2} d x + \int - 7 d x$

Step 11

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

$3 (\frac{1}{2} x^{2} + C) + 4 (- x^{- 1} + C) + \int - 7 d x$

Step 12

Apply the constant rule.

$3 (\frac{1}{2} x^{2} + C) + 4 (- x^{- 1} + C) - 7 x + C$

Step 13

Simplify.

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Step 13.1

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

$3 (\frac{x^{2}}{2} + C) + 4 (- x^{- 1} + C) - 7 x + C$

Step 13.2

Simplify.

$\frac{3 x^{2}}{2} - \frac{4}{x} - 7 x + C$

Step 13.3

Reorder terms.

$\frac{3}{2} x^{2} - \frac{4}{x} - 7 x + C$

Step 14

The answer is the antiderivative of the function $f (x) = \frac{3 x^{3} - 7 x^{2} + 4}{x^{2}}$ .

$F (x) =$ $\frac{3}{2} x^{2} - \frac{4}{x} - 7 x + C$