Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Since $\frac{5}{3}$ is constant with respect to $x$ , move $\frac{5}{3}$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} - 2 (\frac{1}{2} x^{2}) + C$

Step 9.2

Simplify.

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Step 9.2.1

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

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} - 2 \frac{x^{2}}{2} + C$

Step 9.2.2

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

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} + \frac{- 2 x^{2}}{2} + C$

Step 9.2.3

Cancel the common factor of $- 2$ and $2$ .

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Step 9.2.3.1

Factor $2$ out of $- 2 x^{2}$ .

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} + \frac{2 (- x^{2})}{2} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $2$ .

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} + \frac{2 (- x^{2})}{2 (1)} + C$

Step 9.2.3.2.2

Cancel the common factor.

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} + \frac{2 (- x^{2})}{2 \cdot 1} + C$

Step 9.2.3.2.3

Rewrite the expression.

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} + \frac{- x^{2}}{1} + C$

Step 9.2.3.2.4

Divide $- x^{2}$ by $1$ .

$\frac{x^{5}}{10} + \frac{5 x^{4}}{12} - x^{2} + C$

Step 9.3

Reorder terms.

$\frac{1}{10} x^{5} + \frac{5}{12} x^{4} - x^{2} + C$