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

The integral of sinh(x)sin(x) is -(cos(x)sinh(x))/2+(sin(x)cosh(x))/2+C

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integral of sinh(x)sin(x)

\int sinh (x) sin (x) d x

- \frac{cos ( x ) sinh ( x )}{2} + \frac{sin ( x ) cosh ( x )}{2} + C

Solution steps

\int sinh (x) sin (x) d x

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

= - sinh (x) cos (x) - (- (cosh (x) sin (x) - \int sinh (x) sin (x) d x))

Therefore

\int sinh (x) sin (x) d x = - sinh (x) cos (x) - (- (cosh (x) sin (x) - \int sinh (x) sin (x) d x))

Isolate

\int sinh (x) sin (x) d x

= - \frac{cos ( x ) sinh ( x )}{2} + \frac{sin ( x ) cosh ( x )}{2}

Add a constant to the solution

= - \frac{cos ( x ) sinh ( x )}{2} + \frac{sin ( x ) cosh ( x )}{2} + C

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