Calculus Examples

$\frac{d^{2} y}{d x^{2}} = 72 x^{2}$

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 $x$ .

$\frac{d y}{d x} = \int 72 x^{2} d x$

Step 1.2

Since $72$ is constant with respect to $x$ , move $72$ out of the integral.

$\frac{d y}{d x} = 72 \int x^{2} d x$

Step 1.3

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

$\frac{d y}{d x} = 72 (\frac{1}{3} x^{3} + C_{1})$

Step 1.4

Simplify the answer.

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Step 1.4.1

Rewrite $72 (\frac{1}{3} x^{3} + C_{1})$ as $72 (\frac{1}{3}) x^{3} + C_{1}$ .

$\frac{d y}{d x} = 72 (\frac{1}{3}) x^{3} + C_{1}$

Step 1.4.2

Simplify.

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Step 1.4.2.1

Combine $72$ and $\frac{1}{3}$ .

$\frac{d y}{d x} = \frac{72}{3} x^{3} + C_{1}$

Step 1.4.2.2

Cancel the common factor of $72$ and $3$ .

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Step 1.4.2.2.1

Factor $3$ out of $72$ .

$\frac{d y}{d x} = \frac{3 \cdot 24}{3} x^{3} + C_{1}$

Step 1.4.2.2.2

Cancel the common factors.

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Step 1.4.2.2.2.1

Factor $3$ out of $3$ .

$\frac{d y}{d x} = \frac{3 \cdot 24}{3 (1)} x^{3} + C_{1}$

Step 1.4.2.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{3 \cdot 24}{3 \cdot 1} x^{3} + C_{1}$

Step 1.4.2.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{24}{1} x^{3} + C_{1}$

Step 1.4.2.2.2.4

Divide $24$ by $1$ .

$\frac{d y}{d x} = 24 x^{3} + C_{1}$

Step 2

Rewrite the equation.

$d y = (24 x^{3} + C_{1}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int 24 x^{3} + C_{1} d x$

Step 3.2

Apply the constant rule.

$y + C_{2} = \int 24 x^{3} + C_{1} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{2} = \int 24 x^{3} d x + \int C_{1} d x$

Step 3.3.2

Since $24$ is constant with respect to $x$ , move $24$ out of the integral.

$y + C_{2} = 24 \int x^{3} d x + \int C_{1} d x$

Step 3.3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$y + C_{2} = 24 (\frac{1}{4} x^{4} + C_{3}) + \int C_{1} d x$

Step 3.3.4

Apply the constant rule.

$y + C_{2} = 24 (\frac{1}{4} x^{4} + C_{3}) + C_{1} x + C_{4}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y + C_{2} = 24 (\frac{x^{4}}{4} + C_{3}) + C_{1} x + C_{4}$

Step 3.3.5.2

Simplify.

$y + C_{2} = 6 x^{4} + C_{1} x + C_{5}$

Step 3.4

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

$y = 6 x^{4} + C x + D$