What is the integral of 1/(sqrt(2x))cos(sqrt(2x)) ?

The integral of 1/(sqrt(2x))cos(sqrt(2x)) is sin(sqrt(2)sqrt(x))+C

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integral of 1/(sqrt(2x))cos(sqrt(2x))

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

sin (2 x) + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

= \frac{1}{2} \cdot 2 sin (v)

Substitute back

= \frac{1}{2} \cdot 2 sin (2 x)

Simplify

\frac{1}{2} \cdot 2 sin (2 x) : sin (2 x)

= sin (2 x)

Add a constant to the solution

= sin (2 x) + C

x \to 8 - lim (\frac{x - 9}{x - 8})

3 x, 7 x^{2}

\frac{d}{d x} (3^{x^{4} - 2 x^{2}})

x \to 0 lim ((5 x - 3)^{2})

\frac{( 2 s + 1 )}{( s + 1 ) ^{2}}

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