What is the integral of cos((2y)/x) ?

The integral of cos((2y)/x) is xcos((2y)/x)+2y\Si((2y)/x)+C

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integral of cos((2y)/x)

\int cos (\frac{2 y}{x}) d x

x cos (\frac{2 y}{x}) + 2 y Si (\frac{2 y}{x}) + C

Solution steps

\int cos (\frac{2 y}{x}) d x

Apply u-substitution

: \int - \frac{2 y cos ( u )}{u ^{2}} d u

= \int - \frac{2 y cos ( u )}{u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 y \cdot \int \frac{cos ( u )}{u ^{2}} d u

Apply Integration By Parts

: - \frac{cos ( u )}{u} - \int \frac{sin ( u )}{u} d u

= - 2 y (- \frac{cos ( u )}{u} - \int \frac{sin ( u )}{u} d u)

\int \frac{sin ( u )}{u} d u = Si (u)

= - 2 y (- \frac{cos ( u )}{u} - Si (u))

Substitute back

u = \frac{2 y}{x}

= - 2 y (- \frac{cos ( \frac{2 y}{x} )}{\frac{2 y}{x}} - Si (\frac{2 y}{x}))

Simplify

- 2 y (- \frac{cos ( \frac{2 y}{x} )}{\frac{2 y}{x}} - Si (\frac{2 y}{x})) : x cos (\frac{2 y}{x}) + 2 y Si (\frac{2 y}{x})

= x cos (\frac{2 y}{x}) + 2 y Si (\frac{2 y}{x})

Add a constant to the solution

= x cos (\frac{2 y}{x}) + 2 y Si (\frac{2 y}{x}) + C

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What is the integral of cos((2y)/x) ?
The integral of cos((2y)/x) is xcos((2y)/x)+2y\Si((2y)/x)+C