Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

Combine $6$ and $\frac{x^{2}}{2}$ .

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

Step 8.2.3

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

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Step 8.2.3.1

Factor $2$ out of $6 x^{2}$ .

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

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $2$ out of $2$ .

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

Step 8.2.3.2.2

Cancel the common factor.

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

Step 8.2.3.2.3

Rewrite the expression.

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

Step 8.2.3.2.4

Divide $3 x^{2}$ by $1$ .

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

Step 8.3

Reorder terms.

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