Calculus Examples

$\frac{d y}{d x} - \frac{1}{x} = x \cos (x)$

Step 1

Separate the variables.

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Step 1.1

Add $\frac{1}{x}$ to both sides of the equation.

$\frac{d y}{d x} = x \cos (x) + \frac{1}{x}$

Step 1.2

Rewrite the equation.

$d y = (x \cos (x) + \frac{1}{x}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x \cos (x) + \frac{1}{x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x \cos (x) + \frac{1}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int x \cos (x) d x + \int \frac{1}{x} d x$

Step 2.3.2

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

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

Step 2.3.3

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

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

Step 2.3.4

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$y + C_{1} = x \sin (x) - (- \cos (x) + C_{2}) + \ln (| x |) + C_{3}$

Step 2.3.5

Simplify.

$y + C_{1} = x \sin (x) + \cos (x) + \ln (| x |) + C_{4}$

Step 2.3.6

Reorder terms.

$y + C_{1} = x \sin (x) + \ln (| x |) + \cos (x) + C_{4}$

Step 2.4

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

$y = x \sin (x) + \ln (| x |) + \cos (x) + C$