Calculus Examples

$\frac{d y}{d x} = 12 x$

Step 1

Rewrite the equation.

$d y = 12 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 12 x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 12 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$y + C_{1} = 12 \int x d x$

Step 2.3.2

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

$y + C_{1} = 12 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $12 (\frac{1}{2} x^{2} + C_{2})$ as $12 (\frac{1}{2}) x^{2} + C_{2}$ .

$y + C_{1} = 12 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $12$ and $\frac{1}{2}$ .

$y + C_{1} = \frac{12}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $12$ and $2$ .

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Step 2.3.3.2.2.1

Factor $2$ out of $12$ .

$y + C_{1} = \frac{2 \cdot 6}{2} x^{2} + C_{2}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot 6}{2 (1)} x^{2} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot 6}{2 \cdot 1} x^{2} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{6}{1} x^{2} + C_{2}$

Step 2.3.3.2.2.2.4

Divide $6$ by $1$ .

$y + C_{1} = 6 x^{2} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = 6 x^{2} + K$