Calculus Examples

$\frac{d y}{d x} - 15 x^{2} = 8$

Step 1

Separate the variables.

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Step 1.1

Add $15 x^{2}$ to both sides of the equation.

$\frac{d y}{d x} = 8 + 15 x^{2}$

Step 1.2

Rewrite the equation.

$d y = (8 + 15 x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 8 + 15 x^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 8 + 15 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 8 d x + \int 15 x^{2} d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = 8 x + C_{2} + \int 15 x^{2} d x$

Step 2.3.3

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$y + C_{1} = 8 x + C_{2} + 15 \int x^{2} d x$

Step 2.3.4

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

$y + C_{1} = 8 x + C_{2} + 15 (\frac{1}{3} x^{3} + C_{3})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

$y + C_{1} = 8 x + 15 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Combine $15$ and $\frac{1}{3}$ .

$y + C_{1} = 8 x + \frac{15}{3} x^{3} + C_{4}$

Step 2.3.5.2.2

Cancel the common factor of $15$ and $3$ .

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Step 2.3.5.2.2.1

Factor $3$ out of $15$ .

$y + C_{1} = 8 x + \frac{3 \cdot 5}{3} x^{3} + C_{4}$

Step 2.3.5.2.2.2

Cancel the common factors.

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Step 2.3.5.2.2.2.1

Factor $3$ out of $3$ .

$y + C_{1} = 8 x + \frac{3 \cdot 5}{3 (1)} x^{3} + C_{4}$

Step 2.3.5.2.2.2.2

Cancel the common factor.

$y + C_{1} = 8 x + \frac{3 \cdot 5}{3 \cdot 1} x^{3} + C_{4}$

Step 2.3.5.2.2.2.3

Rewrite the expression.

$y + C_{1} = 8 x + \frac{5}{1} x^{3} + C_{4}$

Step 2.3.5.2.2.2.4

Divide $5$ by $1$ .

$y + C_{1} = 8 x + 5 x^{3} + C_{4}$

Step 2.3.6

Reorder terms.

$y + C_{1} = 5 x^{3} + 8 x + C_{4}$

Step 2.4

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

$y = 5 x^{3} + 8 x + K$