Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

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

Step 2.3.6.2

Simplify.

$y + C_{1} = 2 x - \frac{1}{2} x^{2} + \frac{1}{4} x^{4} + C_{5}$

Step 2.3.7

Reorder terms.

$y + C_{1} = - \frac{1}{2} x^{2} + \frac{1}{4} x^{4} + 2 x + C_{5}$

Step 2.4

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

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