Calculus Examples

$\frac{d y}{d x} = 2 x^{4} + 5 x^{3} + x - 4$

Step 1

Rewrite the equation.

$d y = (2 x^{4} + 5 x^{3} + x - 4) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 2 x^{4} + 5 x^{3} + x - 4 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 2 x^{4} + 5 x^{3} + x - 4 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 2 x^{4} d x + \int 5 x^{3} d x + \int x d x + \int - 4 d x$

Step 2.3.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = 2 \int x^{4} d x + \int 5 x^{3} d x + \int x d x + \int - 4 d x$

Step 2.3.3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$y + C_{1} = 2 (\frac{1}{5} x^{5} + C_{2}) + \int 5 x^{3} d x + \int x d x + \int - 4 d x$

Step 2.3.4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$y + C_{1} = 2 (\frac{1}{5} x^{5} + C_{2}) + 5 \int x^{3} d x + \int x d x + \int - 4 d x$

Step 2.3.5

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

$y + C_{1} = 2 (\frac{1}{5} x^{5} + C_{2}) + 5 (\frac{1}{4} x^{4} + C_{3}) + \int x d x + \int - 4 d x$

Step 2.3.6

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

$y + C_{1} = 2 (\frac{1}{5} x^{5} + C_{2}) + 5 (\frac{1}{4} x^{4} + C_{3}) + \frac{1}{2} x^{2} + C_{4} + \int - 4 d x$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$y + C_{1} = 2 (\frac{x^{5}}{5} + C_{2}) + 5 (\frac{1}{4} x^{4} + C_{3}) + \frac{1}{2} x^{2} + C_{4} + \int - 4 d x$

Step 2.3.7.2

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

$y + C_{1} = 2 (\frac{x^{5}}{5} + C_{2}) + 5 (\frac{x^{4}}{4} + C_{3}) + \frac{1}{2} x^{2} + C_{4} + \int - 4 d x$

Step 2.3.8

Apply the constant rule.

$y + C_{1} = 2 (\frac{x^{5}}{5} + C_{2}) + 5 (\frac{x^{4}}{4} + C_{3}) + \frac{1}{2} x^{2} + C_{4} - 4 x + C_{5}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Simplify.

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

Step 2.3.9.2

Reorder terms.

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

Step 2.4

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

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