What is the integral of sin(ln(sqrt(x))) ?

The integral of sin(ln(sqrt(x))) is 2(-1/5 xcos(1/2 ln(x))+2/5 xsin(1/2 ln(x)))+C

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integral of sin(ln(sqrt(x)))

\int sin (ln (x)) d x

2 (- \frac{1}{5} x cos (\frac{1}{2} ln (x)) + \frac{2}{5} x sin (\frac{1}{2} ln (x))) + C

Solution steps

\int sin (ln (x)) d x

Apply u-substitution

: \int sin (ln (u)) \cdot 2 u d u

= \int sin (ln (u)) \cdot 2 u d u

Take the constant out:

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

= 2 \cdot \int sin (ln (u)) u d u

Apply u-substitution

: \int e^{2 v} sin (v) d v

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

: e^{2 v} sin (v) - \int e^{2 v} \cdot 2 sin (v) d v

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

Take the constant out:

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

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

Therefore

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

Isolate

\int e^{2 v} sin (v) d v

= 2 (- \frac{e ^{2 v} cos ( v )}{5} + \frac{2 e ^{2 v} sin ( v )}{5})

Substitute back

= 2 (- \frac{e ^{2 l n (x)} cos ( ln ( x ) )}{5} + \frac{2 e ^{2 l n (x)} sin ( ln ( x ) )}{5})

Simplify

2 (- \frac{e ^{2 l n (x)} cos ( ln ( x ) )}{5} + \frac{2 e ^{2 l n (x)} sin ( ln ( x ) )}{5}) : 2 (- \frac{1}{5} x cos (\frac{1}{2} ln (x)) + \frac{2}{5} x sin (\frac{1}{2} ln (x)))

= 2 (- \frac{1}{5} x cos (\frac{1}{2} ln (x)) + \frac{2}{5} x sin (\frac{1}{2} ln (x)))

Add a constant to the solution

= 2 (- \frac{1}{5} x cos (\frac{1}{2} ln (x)) + \frac{2}{5} x sin (\frac{1}{2} ln (x))) + C

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What is the integral of sin(ln(sqrt(x))) ?
The integral of sin(ln(sqrt(x))) is 2(-1/5 xcos(1/2 ln(x))+2/5 xsin(1/2 ln(x)))+C