Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

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Step 2.3.4.1

Simplify.

$y + C_{1} = - \frac{1}{2} \cdot \frac{1}{x^{2}} + 2 x + C_{4}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

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

$y + C_{1} = - \frac{1}{x^{2} \cdot 2} + 2 x + C_{4}$

Step 2.3.4.2.2

Move $2$ to the left of $x^{2}$ .

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

Step 2.3.5

Reorder terms.

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

Step 2.4

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

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