Calculus Examples

$\int \frac{3 x^{5} + 2}{4 x^{2}} d x$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{3 x^{5} + 2}{x^{2}} d x$

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.

$\frac{1}{4} \int (3 x^{5} + 2) {(x^{2})}^{- 1} d x$

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

$\frac{1}{4} \int (3 x^{5} + 2) x^{2 \cdot - 1} d x$

Step 2.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{4} \int (3 x^{5} + 2) x^{- 2} d x$

Step 3

Multiply $(3 x^{5} + 2) x^{- 2}$ .

$\frac{1}{4} \int 3 x^{5} x^{- 2} + 2 x^{- 2} d x$

Step 4

Multiply $x^{5}$ by $x^{- 2}$ by adding the exponents.

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Step 4.1

Move $x^{- 2}$ .

$\frac{1}{4} \int 3 (x^{- 2} x^{5}) + 2 x^{- 2} d x$

Step 4.2

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

$\frac{1}{4} \int 3 x^{- 2 + 5} + 2 x^{- 2} d x$

Step 4.3

Add $- 2$ and $5$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Reorder terms.

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