What is the integral of (cos(u))/(sqrt(u)) ?

The integral of (cos(u))/(sqrt(u)) is sqrt(2)sqrt(pi)\C(sqrt(2/pi)sqrt(u))+C

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integral of (cos(u))/(sqrt(u))

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

2 π C (\frac{2}{π} u) + C

Solution steps

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

Apply u-substitution

= \int 2 cos (v^{2}) d v

Take the constant out:

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

= 2 \cdot \int cos (v^{2}) d v

Use the nonelementary integral:

\int cos (x^{2}) d x = \frac{π}{2} C (\frac{2}{π} x)

= 2 \frac{π}{2} C (\frac{2}{π} v)

Substitute back

v = u

= 2 \frac{π}{2} C (\frac{2}{π} u)

Simplify

2 \frac{π}{2} C (\frac{2}{π} u) : 2 π C (\frac{2}{π} u)

= 2 π C (\frac{2}{π} u)

Add a constant to the solution

= 2 π C (\frac{2}{π} u) + C

What is the integral of (cos(u))/(sqrt(u)) ?
The integral of (cos(u))/(sqrt(u)) is sqrt(2)sqrt(pi)\C(sqrt(2/pi)sqrt(u))+C