What is the integral of cos(xy^{-1/2}) ?

The integral of cos(xy^{-1/2}) is y^{1/2}sin(x/(y^{1/2)})+C

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integral of cos(xy^{-1/2})

\int cos (x y^{- \frac{1}{2}}) d x

y^{\frac{1}{2}} sin (\frac{x}{y ^{\frac{1}{2}}}) + C

Solution steps

\int cos (x y^{- \frac{1}{2}}) d x

Apply u-substitution

: \int cos (u) y^{\frac{1}{2}} d u

= \int cos (u) y^{\frac{1}{2}} d u

Take the constant out:

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

= y^{\frac{1}{2}} \cdot \int cos (u) d u

Use the common integral:

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

= y^{\frac{1}{2}} sin (u)

Substitute back

u = y^{- \frac{1}{2}} x

= y^{\frac{1}{2}} sin (y^{- \frac{1}{2}} x)

Multiply

y^{- \frac{1}{2}} x : \frac{x}{y ^{\frac{1}{2}}}

= y^{\frac{1}{2}} sin (\frac{x}{y ^{\frac{1}{2}}})

Add a constant to the solution

= y^{\frac{1}{2}} sin (\frac{x}{y ^{\frac{1}{2}}}) + C

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What is the integral of cos(xy^{-1/2}) ?
The integral of cos(xy^{-1/2}) is y^{1/2}sin(x/(y^{1/2)})+C