Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.3

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

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

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $3 (- x^{- 1} + C_{2})$ as $3 (- \frac{1}{x}) + C_{2}$ .

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

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $- 1$ by $3$ .

$y + C_{1} = - 3 \frac{1}{x} + C_{2}$

Step 2.3.4.2.2

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

$y + C_{1} = \frac{- 3}{x} + C_{2}$

Step 2.3.4.2.3

Move the negative in front of the fraction.

$y + C_{1} = - \frac{3}{x} + C_{2}$

Step 2.4

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

$y = - \frac{3}{x} + K$