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

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

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

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

π C (\frac{2}{π} x) + C

Solution steps

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

2 x = 2 x

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

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 ( x )}{x} d x

Apply u-substitution

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

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

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 (u^{2}) d u

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} u)

Substitute back

u = x

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

Simplify

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

= π C (\frac{2}{π} x)

Add a constant to the solution

= π C (\frac{2}{π} x) + C

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