Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 3 x^{2} + 1 d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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

Step 2.4

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

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