What is the integral of (ycos(xy)) ?

The integral of (ycos(xy)) is sin(yx)+C

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integral of (ycos(xy))

\int (y cos (x y)) d x

sin (y x) + C

Solution steps

\int y cos (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 (y x) d x

Apply u-substitution

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

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

Take the constant out:

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

= y \frac{1}{y} \cdot \int cos (u) d u

Use the common integral:

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

= y \frac{1}{y} sin (u)

Substitute back

u = y x

= y \frac{1}{y} sin (y x)

Simplify

y \frac{1}{y} sin (y x) : sin (y x)

= sin (y x)

Add a constant to the solution

= sin (y x) + C

x \to - 7 + lim (\frac{x + 8}{x + 7})

\frac{d}{d x} (5 ln (4 x^{2}))

x^{8} + y^{7} = 9

y^{^{'}} + 3 x y^{3} = 0

f (x) = \frac{x + 3}{x - 3}