Calculus Examples

$\int \frac{16 x^{4} - 3 x^{3} + 8 x}{x^{3}} d x$

Step 1

Simplify.

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Step 1.1

Factor $x$ out of $16 x^{4} - 3 x^{3} + 8 x$ .

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Step 1.1.1

Factor $x$ out of $16 x^{4}$ .

$\int \frac{x (16 x^{3}) - 3 x^{3} + 8 x}{x^{3}} d x$

Step 1.1.2

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

$\int \frac{x (16 x^{3}) + x (- 3 x^{2}) + 8 x}{x^{3}} d x$

Step 1.1.3

Factor $x$ out of $8 x$ .

$\int \frac{x (16 x^{3}) + x (- 3 x^{2}) + x \cdot 8}{x^{3}} d x$

Step 1.1.4

Factor $x$ out of $x (16 x^{3}) + x (- 3 x^{2})$ .

$\int \frac{x (16 x^{3} - 3 x^{2}) + x \cdot 8}{x^{3}} d x$

Step 1.1.5

Factor $x$ out of $x (16 x^{3} - 3 x^{2}) + x \cdot 8$ .

$\int \frac{x (16 x^{3} - 3 x^{2} + 8)}{x^{3}} d x$

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

$\int \frac{x (16 x^{3} - 3 x^{2} + 8)}{x \cdot x^{2}} d x$

Step 1.2.2

Cancel the common factor.

$\int \frac{x (16 x^{3} - 3 x^{2} + 8)}{x \cdot x^{2}} d x$

Step 1.2.3

Rewrite the expression.

$\int \frac{16 x^{3} - 3 x^{2} + 8}{x^{2}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\int (16 x^{3} - 3 x^{2} + 8) {(x^{2})}^{- 1} d x$

Step 2.2

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

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Step 2.2.1

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

$\int (16 x^{3} - 3 x^{2} + 8) x^{2 \cdot - 1} d x$

Step 2.2.2

Multiply $2$ by $- 1$ .

$\int (16 x^{3} - 3 x^{2} + 8) x^{- 2} d x$

Step 3

Expand $(16 x^{3} - 3 x^{2} + 8) x^{- 2}$ .

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Step 3.1

Apply the distributive property.

$\int (16 x^{3} - 3 x^{2}) x^{- 2} + 8 x^{- 2} d x$

Step 3.2

Apply the distributive property.

$\int 16 x^{3} x^{- 2} - 3 x^{2} x^{- 2} + 8 x^{- 2} d x$

Step 3.3

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

$\int 16 x^{3 - 2} - 3 x^{2} x^{- 2} + 8 x^{- 2} d x$

Step 3.4

Subtract $2$ from $3$ .

$\int 16 x^{1} - 3 x^{2} x^{- 2} + 8 x^{- 2} d x$

Step 3.5

Simplify.

$\int 16 x - 3 x^{2} x^{- 2} + 8 x^{- 2} d x$

Step 3.6

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

$\int 16 x - 3 x^{2 - 2} + 8 x^{- 2} d x$

Step 3.7

Subtract $2$ from $2$ .

$\int 16 x - 3 x^{0} + 8 x^{- 2} d x$

Step 3.8

Anything raised to $0$ is $1$ .

$\int 16 x - 3 \cdot 1 + 8 x^{- 2} d x$

Step 3.9

Multiply $- 3$ by $1$ .

$\int 16 x - 3 + 8 x^{- 2} d x$

Step 3.10

Move $- 3$ .

$\int 16 x + 8 x^{- 2} - 3 d x$

Step 4

Split the single integral into multiple integrals.

$\int 16 x d x + \int 8 x^{- 2} d x + \int - 3 d x$

Step 5

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int x d x + \int 8 x^{- 2} d x + \int - 3 d x$

Step 6

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

$16 (\frac{1}{2} x^{2} + C) + \int 8 x^{- 2} d x + \int - 3 d x$

Step 7

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$16 (\frac{1}{2} x^{2} + C) + 8 \int x^{- 2} d x + \int - 3 d x$

Step 8

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

$16 (\frac{1}{2} x^{2} + C) + 8 (- x^{- 1} + C) + \int - 3 d x$

Step 9

Apply the constant rule.

$16 (\frac{1}{2} x^{2} + C) + 8 (- x^{- 1} + C) - 3 x + C$

Step 10

Simplify.

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Step 10.1

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

$16 (\frac{x^{2}}{2} + C) + 8 (- x^{- 1} + C) - 3 x + C$

Step 10.2

Simplify.

$8 x^{2} - \frac{8}{x} - 3 x + C$