What is the integral of e^x*sin(x-pi) ?

The integral of e^x*sin(x-pi) is -1/2 e^xsin(x)-1/2 e^xcos(x)+e^xcos(x)+C

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integral of e^x*sin(x-pi)

\int e^{x} \cdot sin (x - π) d x

- \frac{1}{2} e^{x} sin (x) - \frac{1}{2} e^{x} cos (x) + e^{x} cos (x) + C

Solution steps

\int e^{x} sin (x - π) d x

Apply Integration By Parts

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

Apply Integration By Parts

= e^{x} cos (x) - (e^{x} sin (x) - \int e^{x} sin (x) d x)

Apply Integration By Parts

= e^{x} cos (x) - (e^{x} sin (x) - (- e^{x} cos (x) - \int - e^{x} cos (x) d x))

Take the constant out:

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

= e^{x} cos (x) - (e^{x} sin (x) - (- e^{x} cos (x) - (- \int e^{x} cos (x) d x)))

Therefore

- \int e^{x} cos (x) d x + cos (x) e^{x} = e^{x} cos (x) - (e^{x} sin (x) - (- e^{x} cos (x) - (- \int e^{x} cos (x) d x)))

Isolate

\int e^{x} cos (x) d x

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{2} e^{x} sin (x) - \frac{1}{2} e^{x} cos (x) + e^{x} cos (x) + C

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