What is the integral of ycos(x/y) ?

The integral of ycos(x/y) is y^2sin(x/y)+C

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integral of ycos(x/y)

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

y^{2} sin (\frac{x}{y}) + C

Solution steps

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

Take the constant out:

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

= y \cdot \int cos (\frac{x}{y}) d x

Apply u-substitution

: \int cos (u) y d u

= y \cdot \int cos (u) y d u

Take the constant out:

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

= yy \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= yy sin (u)

Substitute back

u = \frac{x}{y}

= yy sin (\frac{x}{y})

Simplify

yy sin (\frac{x}{y}) : y^{2} sin (\frac{x}{y})

= y^{2} sin (\frac{x}{y})

Add a constant to the solution

= y^{2} sin (\frac{x}{y}) + C

f (x) = 2 (sin^{3} (x))

\int \frac{sin ( 2 x )}{14 + cos ^{2} ( x )} d x

y^{^{'}} + \frac{2}{x} y = y^{2}, y (1) = 3

\frac{d}{d x} (\frac{a}{2 y})

t^{3} y^{^{'}} + 4 t^{2} y = 0