What is the integral of 5/(sqrt(2x))sin(sqrt(2x)) ?

The integral of 5/(sqrt(2x))sin(sqrt(2x)) is -5cos(sqrt(2)sqrt(x))+C

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integral of 5/(sqrt(2x))sin(sqrt(2x))

\int \frac{5}{2 x} sin (2 x) d x

- 5 cos (2 x) + C

Solution steps

\int \frac{5}{2 x} sin (2 x) d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{2 x} sin (2 x) d x

Apply u-substitution

: \int \frac{sin ( u )}{2 u} d u

= 5 \cdot \int \frac{sin ( u )}{2 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{sin ( u )}{u} d u

Apply u-substitution

: \int 2 sin (v) d v

= 5 \cdot \frac{1}{2} \cdot \int 2 sin (v) d v

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot 2 \cdot \int sin (v) d v

Use the common integral:

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

= 5 \cdot \frac{1}{2} \cdot 2 (- cos (v))

Substitute back

= 5 \cdot \frac{1}{2} \cdot 2 (- cos (2 x))

Simplify

5 \cdot \frac{1}{2} \cdot 2 (- cos (2 x)) : - 5 cos (2 x)

= - 5 cos (2 x)

Add a constant to the solution

= - 5 cos (2 x) + C

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What is the integral of 5/(sqrt(2x))sin(sqrt(2x)) ?
The integral of 5/(sqrt(2x))sin(sqrt(2x)) is -5cos(sqrt(2)sqrt(x))+C