Calculus Examples

$\frac{d y}{d x} = 5 + x$

Step 1

Rewrite the equation.

$d y = (5 + x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 5 + x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 5 + x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 5 d x + \int x d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = 5 x + C_{2} + \int x d x$

Step 2.3.3

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

$y + C_{1} = 5 x + C_{2} + \frac{1}{2} x^{2} + C_{3}$

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

Step 2.4

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

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