Calculus Examples

$(y^{2} - y) \frac{d y}{d x} = x + \ln (x)$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} - y d y = \int x + \ln (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y^{2} d y + \int - y d y = \int x + \ln (x) d x$

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

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

Step 2.3.4

Simplify.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Cancel the common factor of $x$ .

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Step 2.3.4.2.1

Cancel the common factor.

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

Step 2.3.4.2.2

Rewrite the expression.

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

Simplify.

$\frac{1}{3} y^{3} - \frac{1}{2} y^{2} + C_{3} = \frac{1}{2} x^{2} + \ln (x) x - x + C_{6}$

Step 2.3.7

Reorder terms.

$\frac{1}{3} y^{3} - \frac{1}{2} y^{2} + C_{3} = \frac{1}{2} x^{2} + x \ln (x) - x + C_{6}$

Step 2.4

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

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