Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\int (3 x^{4} - 6) {(x^{2})}^{- 1} d x$

Step 1.2

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

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Step 1.2.1

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

$\int (3 x^{4} - 6) x^{2 \cdot - 1} d x$

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

Multiply $(3 x^{4} - 6) x^{- 2}$ .

$\int 3 x^{4} x^{- 2} - 6 x^{- 2} d x$

Step 3

Multiply $x^{4}$ by $x^{- 2}$ by adding the exponents.

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Step 3.1

Move $x^{- 2}$ .

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

Step 3.2

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

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

Step 3.3

Add $- 2$ and $4$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

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

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

Step 9.2.2

Cancel the common factor of $3$ .

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Step 9.2.2.1

Cancel the common factor.

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

Step 9.2.2.2

Rewrite the expression.

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

Step 9.2.3

Multiply $x^{3}$ by $1$ .

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

Step 9.2.4

Multiply $- 1$ by $- 6$ .

$x^{3} + 6 x^{- 1} + C$