Calculus Examples

$\int 4 x^{2} - 6 x d x$

Step 1

Split the single integral into multiple integrals.

$\int 4 x^{2} d x + \int - 6 x d x$

Step 2

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

$4 \int x^{2} d x + \int - 6 x d x$

Step 3

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

$4 (\frac{1}{3} x^{3} + C) + \int - 6 x d x$

Step 4

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

$4 (\frac{1}{3} x^{3} + C) - 6 \int x d x$

Step 5

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

$4 (\frac{1}{3} x^{3} + C) - 6 (\frac{1}{2} x^{2} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$4 (\frac{1}{3}) x^{3} - 6 (\frac{1}{2}) x^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} x^{3} - 6 (\frac{1}{2}) x^{2} + C$

Step 6.2.2

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

$\frac{4}{3} x^{3} + \frac{- 6}{2} x^{2} + C$

Step 6.2.3

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

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Step 6.2.3.1

Factor $2$ out of $- 6$ .

$\frac{4}{3} x^{3} + \frac{2 \cdot - 3}{2} x^{2} + C$

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $2$ out of $2$ .

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

Step 6.2.3.2.2

Cancel the common factor.

$\frac{4}{3} x^{3} + \frac{2 \cdot - 3}{2 \cdot 1} x^{2} + C$

Step 6.2.3.2.3

Rewrite the expression.

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

Step 6.2.3.2.4

Divide $- 3$ by $1$ .

$\frac{4}{3} x^{3} - 3 x^{2} + C$