Integral of exp(-x)sinx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = - x$.

      Then let $du = - dx$ and substitute $du$:

      $$\int e^{u} \sin{\left(u \right)}\, du$$

      1. Use integration by parts, noting that the integrand eventually repeats itself.

         1. For the integrand $e^{u} \sin{\left(u \right)}$:

            Let $u{\left(u \right)} = \sin{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

            Then $\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - \int e^{u} \cos{\left(u \right)}\, du$.

         2. For the integrand $e^{u} \cos{\left(u \right)}$:

            Let $u{\left(u \right)} = \cos{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

            Then $\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \sin{\left(u \right)}\right)\, du$.

         3. Notice that the integrand has repeated itself, so move it to one side:

            $$2 \int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)}$$

            Therefore,

            $$\int e^{u} \sin{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}$$

      Now substitute $u$ back in:

      $$- \frac{e^{- x} \sin{\left(x \right)}}{2} - \frac{e^{- x} \cos{\left(x \right)}}{2}$$

   Method #2

   1. Use integration by parts, noting that the integrand eventually repeats itself.

      1. For the integrand $e^{- x} \sin{\left(x \right)}$:

         Let $u{\left(x \right)} = \sin{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- x}$.

         Then $\int e^{- x} \sin{\left(x \right)}\, dx = - \int \left(- e^{- x} \cos{\left(x \right)}\right)\, dx - e^{- x} \sin{\left(x \right)}$.

      2. For the integrand $- e^{- x} \cos{\left(x \right)}$:

         Let $u{\left(x \right)} = - \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- x}$.

         Then $\int e^{- x} \sin{\left(x \right)}\, dx = \int \left(- e^{- x} \sin{\left(x \right)}\right)\, dx - e^{- x} \sin{\left(x \right)} - e^{- x} \cos{\left(x \right)}$.

      3. Notice that the integrand has repeated itself, so move it to one side:

         $$2 \int e^{- x} \sin{\left(x \right)}\, dx = - e^{- x} \sin{\left(x \right)} - e^{- x} \cos{\left(x \right)}$$

         Therefore,

         $$\int e^{- x} \sin{\left(x \right)}\, dx = - \frac{e^{- x} \sin{\left(x \right)}}{2} - \frac{e^{- x} \cos{\left(x \right)}}{2}$$

2. Now simplify:

   $$- \frac{\sqrt{2} e^{- x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}$$

3. Add the constant of integration:

   $$- \frac{\sqrt{2} e^{- x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$$

The answer is:

$$- \frac{\sqrt{2} e^{- x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$$