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

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

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integral of cos(x^{1/2})

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

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

Solution steps

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

Apply u-substitution

: \int cos (u) \cdot 2 u d u

= \int cos (u) \cdot 2 u d u

Take the constant out:

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

= 2 \cdot \int cos (u) u d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= 2 (u sin (u) - \int sin (u) d u)

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

= 2 (u sin (u) - (- cos (u)))

Substitute back

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

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

Simplify

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

Add a constant to the solution

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

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