What is the integral of (cos(u))/(sqrt(2u)) ?

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

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integral of (cos(u))/(sqrt(2u))

\int \frac{cos ( u )}{2 u} d u

π C (\frac{2}{π} u) + C

Solution steps

\int \frac{cos ( u )}{2 u} d u

2 u = 2 u

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

Take the constant out:

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

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

Apply u-substitution

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

= \frac{1}{2} \cdot \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

= \frac{1}{2} \cdot 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)

= \frac{1}{2} \cdot 2 \frac{π}{2} C (\frac{2}{π} v)

Substitute back

v = u

= \frac{1}{2} \cdot 2 \frac{π}{2} C (\frac{2}{π} u)

Simplify

\frac{1}{2} \cdot 2 \frac{π}{2} C (\frac{2}{π} u) : π C (\frac{2}{π} u)

= π C (\frac{2}{π} u)

Add a constant to the solution

= π C (\frac{2}{π} u) + C

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