Calculus Examples

$\frac{d^{2} y}{d x^{2}} = 12$

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

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

Step 1.2

Apply the constant rule.

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

Step 2

Rewrite the equation.

$d y = (12 x + C_{1}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Apply the constant rule.

$y + C_{2} = \int 12 x + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int 12 x d x + \int C_{1} d x$

Step 3.3.2

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

$y + C_{2} = 12 \int x d x + \int C_{1} d x$

Step 3.3.3

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

$y + C_{2} = 12 (\frac{1}{2} x^{2} + C_{3}) + \int C_{1} d x$

Step 3.3.4

Apply the constant rule.

$y + C_{2} = 12 (\frac{1}{2} x^{2} + C_{3}) + C_{1} x + C_{4}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y + C_{2} = 12 (\frac{x^{2}}{2} + C_{3}) + C_{1} x + C_{4}$

Step 3.3.5.2

Simplify.

$y + C_{2} = 6 x^{2} + C_{1} x + C_{5}$

Step 3.4

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

$y = 6 x^{2} + C x + D$