Calculus Examples

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

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 $x$ .

$\frac{d y}{d x} = \int - \frac{1}{x^{3}} d x$

Step 1.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{d y}{d x} = - \int \frac{1}{x^{3}} d x$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\frac{d y}{d x} = - \int {(x^{3})}^{- 1} d x$

Step 1.3.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d y}{d x} = - \int x^{3 \cdot - 1} d x$

Step 1.3.2.2

Multiply $3$ by $- 1$ .

$\frac{d y}{d x} = - \int x^{- 3} d x$

Step 1.4

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

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

Step 1.5

Simplify the answer.

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Step 1.5.1

Simplify.

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Step 1.5.1.1

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

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

Step 1.5.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.5.2

Simplify.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.3.2

Multiply $\frac{1}{2 x^{2}}$ by $1$ .

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

Step 2

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Apply the constant rule.

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

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

Step 3.3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 3.3.3

Apply basic rules of exponents.

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Step 3.3.3.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.3.3.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 3.3.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{2} = \frac{1}{2} \int x^{2 \cdot - 1} d x + \int C_{1} d x$

Step 3.3.3.2.2

Multiply $2$ by $- 1$ .

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

Step 3.3.4

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

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

Step 3.3.5

Apply the constant rule.

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

Step 3.3.6

Simplify.

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Step 3.3.6.1

Simplify.

$y + C_{2} = \frac{1}{2} (- \frac{1}{x}) + C_{1} x + C_{5}$

Step 3.3.6.2

Multiply $\frac{1}{2}$ by $\frac{1}{x}$ .

$y + C_{2} = - \frac{1}{2 x} + C_{1} x + C_{5}$

Step 3.3.7

Reorder terms.

$y + C_{2} = C_{1} x - \frac{1}{2 x} + C_{5}$

Step 3.4

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

$y = C x - \frac{1}{2 x} + D$