Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

$8 (\frac{1}{4} x^{4} + C) + \int \frac{1}{2 x^{2}} d x$

Step 5

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$8 (\frac{1}{4} x^{4} + C) + \frac{1}{2} \int \frac{1}{x^{2}} d x$

Step 6

Simplify the expression.

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Step 6.1

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

$8 (\frac{1}{4} x^{4} + C) + \frac{1}{2} \int {(x^{2})}^{- 1} d x$

Step 6.2

Simplify.

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Step 6.2.1

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

$8 (\frac{x^{4}}{4} + C) + \frac{1}{2} \int {(x^{2})}^{- 1} d x$

Step 6.2.2

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

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Step 6.2.2.1

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

$8 (\frac{x^{4}}{4} + C) + \frac{1}{2} \int x^{2 \cdot - 1} d x$

Step 6.2.2.2

Multiply $2$ by $- 1$ .

$8 (\frac{x^{4}}{4} + C) + \frac{1}{2} \int x^{- 2} d x$

Step 7

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

$8 (\frac{x^{4}}{4} + C) + \frac{1}{2} (- x^{- 1} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

Move $x^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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