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

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

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integral of sin(ln(x^2))

\int sin (ln (x^{2})) d x

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

Solution steps

\int sin (ln (x^{2})) d x

Simplify

ln (x^{2}) : 2 ln (x)

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

(5 x^{2} + 3 y + 3 y^{2}) d x + (x + 2 x y) d y = 0

\int \frac{1 + x}{1 + x} d x

\frac{d y}{d x} - y = x y^{5}

\int 14 cos^{3} (x) d x

\int (2 x + 1) d x

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