Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

$\frac{1}{5} x^{5} + C - 2 (\frac{x^{4}}{4} + C) - x^{- 1} + C$

Step 7

Simplify.

$\frac{x^{5}}{5} - \frac{x^{4}}{2} - x^{- 1} + C$

Step 8

Reorder terms.

$\frac{1}{5} x^{5} - \frac{1}{2} x^{4} - x^{- 1} + C$