Calculus Examples

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Move $x^{- 2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Add $- 2$ and $2$ .

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

Step 3.2

Simplify $3 x^{0}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

$3 x + C - 4 (- x^{- 1} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $- 1$ by $- 4$ .

$3 x + 4 \frac{1}{x} + C$

Step 8.2.2

Combine $4$ and $\frac{1}{x}$ .

$3 x + \frac{4}{x} + C$