Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify.

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

Step 5.2

Simplify.

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Step 5.2.1

Combine $- 4$ and $\frac{1}{2}$ .

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

Step 5.2.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.2.2.1

Factor $2$ out of $- 4$ .

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

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2.2

Cancel the common factor.

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

Step 5.2.2.2.3

Rewrite the expression.

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

Step 5.2.2.2.4

Divide $- 2$ by $1$ .

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