What is the integral of sinh(ln(x)) ?

The integral of sinh(ln(x)) is xsinh(ln(x))-1/2 (ln|x|+(x^2)/2)+C

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integral of sinh(ln(x))

\int sinh (ln (x)) d x

x sinh (ln (x)) - \frac{1}{2} (ln ∣ x ∣ + \frac{x ^{2}}{2}) + C

Solution steps

\int sinh (ln (x)) d x

Apply Integration By Parts

: x sinh (ln (x)) - \int cosh (ln (x)) d x

= x sinh (ln (x)) - \int cosh (ln (x)) d x

\int cosh (ln (x)) d x = \frac{1}{2} (ln ∣ x ∣ + \frac{x ^{2}}{2})

= x sinh (ln (x)) - \frac{1}{2} (ln ∣ x ∣ + \frac{x ^{2}}{2})

Add a constant to the solution

= x sinh (ln (x)) - \frac{1}{2} (ln ∣ x ∣ + \frac{x ^{2}}{2}) + C

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What is the integral of sinh(ln(x)) ?
The integral of sinh(ln(x)) is xsinh(ln(x))-1/2 (ln|x|+(x^2)/2)+C