Calculus Examples

$6 x d x$

Step 1

Write $6 x d x$ as a function.

$f (x) = 6 x d x$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 6 x d x d x$

Step 4

Simplify.

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Step 4.1

Raise $x$ to the power of $1$ .

$\int 6 d (x^{1} x) d x$

Step 4.2

Raise $x$ to the power of $1$ .

$\int 6 d (x^{1} x^{1}) d x$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 6 d x^{1 + 1} d x$

Step 4.4

Add $1$ and $1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Rewrite $6 d (\frac{1}{3} x^{3} + C)$ as $6 d \frac{1}{3} x^{3} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Combine $6$ and $\frac{1}{3}$ .

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

Step 7.2.2

Combine $d$ and $\frac{6}{3}$ .

$\frac{d \cdot 6}{3} x^{3} + C$

Step 7.2.3

Move $6$ to the left of $d$ .

$\frac{6 \cdot d}{3} x^{3} + C$

Step 7.2.4

Cancel the common factor of $6$ and $3$ .

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Step 7.2.4.1

Factor $3$ out of $6 d$ .

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

Step 7.2.4.2

Cancel the common factors.

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Step 7.2.4.2.1

Factor $3$ out of $3$ .

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

Step 7.2.4.2.2

Cancel the common factor.

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

Step 7.2.4.2.3

Rewrite the expression.

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

Step 7.2.4.2.4

Divide $2 d$ by $1$ .

$2 d x^{3} + C$

Step 8

The answer is the antiderivative of the function $f (x) = 6 x d x$ .

$F (x) =$ $2 d x^{3} + C$