What is the integral of 2sin(ln(x)) ?

The integral of 2sin(ln(x)) is xsin(ln(x))-xcos(ln(x))+C

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integral of 2sin(ln(x))

\int 2 sin (ln (x)) d x

x sin (ln (x)) - x cos (ln (x)) + C

Solution steps

\int 2 sin (ln (x)) d x

Take the constant out:

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

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

Apply Integration By Parts

: x sin (ln (x)) - \int cos (ln (x)) d x

= 2 (x sin (ln (x)) - \int cos (ln (x)) d x)

\int cos (ln (x)) d x = \frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x))

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

Simplify

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

= x sin (ln (x)) - x cos (ln (x))

Add a constant to the solution

= x sin (ln (x)) - x cos (ln (x)) + C

\frac{f ( x )}{g ( x ) + 1}

\frac{x ^{2}}{3 + 8 x}

\int 3 t + 1 d t

\frac{d}{d x} (x^{3} - 3 x^{2} + 5 x - 2)

\frac{d}{d x} (a e^{x} + \frac{b}{x} + \frac{c}{x ^{2}})

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