Calculus Examples

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

Step 1

Rewrite the equation.

$(3 y^{2} + 2) d y = 1 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 3 y^{2} + 2 d y = \int d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 3 y^{2} d y + \int 2 d y = \int d x$

Step 2.2.2

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

$3 \int y^{2} d y + \int 2 d y = \int d x$

Step 2.2.3

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

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

Combine $\frac{1}{3}$ and $y^{3}$ .

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

Step 2.2.5.2

Simplify.

$y^{3} + 2 y + C_{3} = \int d x$

Step 2.3

Apply the constant rule.

$y^{3} + 2 y + C_{3} = x + C_{4}$

Step 2.4

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

$y^{3} + 2 y = x + K$