What is the integral of cos(yx) ?

The integral of cos(yx) is 1/y sin(yx)+C

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integral of cos(yx)

\int cos (y x) d x

\frac{1}{y} sin (y x) + C

Solution steps

\int cos (y x) d x

Apply u-substitution

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

= \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

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

Use the common integral:

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

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

Substitute back

u = y x

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

Add a constant to the solution

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

x \frac{d y}{d x} + 2 y = x^{- 3}

y = e^{2 x} + e^{- 2 x}

\frac{d y}{d x} = x + y, y (0) = 0

f (x) = 3 e^{- 2 x} ln (x)

\frac{d}{d x} (arcsin (5 x^{3}))