Calculus Examples

$\int (3 x^{3} + 6 + 6 x) d x$

Step 1

Remove parentheses.

$\int 3 x^{3} + 6 + 6 x d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 x^{3} d x + \int 6 d x + \int 6 x d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{3} d x + \int 6 d x + \int 6 x d x$

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

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

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

Step 9.3.2

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

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Step 9.3.2.1

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

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

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $2$ out of $2$ .

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

Step 9.3.2.2.2

Cancel the common factor.

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

Step 9.3.2.2.3

Rewrite the expression.

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

Step 9.3.2.2.4

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

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

Step 9.4

Reorder terms.

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