Calculus Examples

$x^{2} - 6 x$

Step 1

Split the single integral into multiple integrals.

$\int x^{2} d x + \int - 6 x d x$

Step 2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3} + C + \int - 6 x d x$

Step 3

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{3} x^{3} + C - 6 \int x d x$

Step 4

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

$\frac{1}{3} x^{3} + C - 6 (\frac{1}{2} x^{2} + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

$\frac{1}{3} x^{3} - 6 (\frac{1}{2}) x^{2} + C$

Step 5.2

Simplify.

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Step 5.2.1

Combine $- 6$ and $\frac{1}{2}$ .

$\frac{1}{3} x^{3} + \frac{- 6}{2} x^{2} + C$

Step 5.2.2

Cancel the common factor of $- 6$ and $2$ .

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Step 5.2.2.1

Factor $2$ out of $- 6$ .

$\frac{1}{3} x^{3} + \frac{2 \cdot - 3}{2} x^{2} + C$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2.2

Cancel the common factor.

$\frac{1}{3} x^{3} + \frac{2 \cdot - 3}{2 \cdot 1} x^{2} + C$

Step 5.2.2.2.3

Rewrite the expression.

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

Step 5.2.2.2.4

Divide $- 3$ by $1$ .

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

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