Calculus Examples

$d y = 18 x^{2} d x$

Step 1

Set up an integral on each side.

$\int d y = \int 18 x^{2} d x$

Step 2

Apply the constant rule.

$y + C_{1} = \int 18 x^{2} d x$

Step 3

Integrate the right side.

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Step 3.1

Since $18$ is constant with respect to $x$ , move $18$ out of the integral.

$y + C_{1} = 18 \int x^{2} d x$

Step 3.2

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

$y + C_{1} = 18 (\frac{1}{3} x^{3} + C_{2})$

Step 3.3

Simplify the answer.

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Step 3.3.1

Rewrite $18 (\frac{1}{3} x^{3} + C_{2})$ as $18 (\frac{1}{3}) x^{3} + C_{2}$ .

$y + C_{1} = 18 (\frac{1}{3}) x^{3} + C_{2}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Combine $18$ and $\frac{1}{3}$ .

$y + C_{1} = \frac{18}{3} x^{3} + C_{2}$

Step 3.3.2.2

Cancel the common factor of $18$ and $3$ .

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Step 3.3.2.2.1

Factor $3$ out of $18$ .

$y + C_{1} = \frac{3 \cdot 6}{3} x^{3} + C_{2}$

Step 3.3.2.2.2

Cancel the common factors.

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Step 3.3.2.2.2.1

Factor $3$ out of $3$ .

$y + C_{1} = \frac{3 \cdot 6}{3 (1)} x^{3} + C_{2}$

Step 3.3.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{3 \cdot 6}{3 \cdot 1} x^{3} + C_{2}$

Step 3.3.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{6}{1} x^{3} + C_{2}$

Step 3.3.2.2.2.4

Divide $6$ by $1$ .

$y + C_{1} = 6 x^{3} + C_{2}$

Step 4

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

$y = 6 x^{3} + K$