Integral of e^x(sinx-cosx)dx dx

1. Rewrite the integrand:

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

2. Integrate term-by-term:

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

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

      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,

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

   1. The integral of a constant times a function is the constant times the integral of the function:

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

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

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

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

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

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

            Therefore,

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

      So, the result is: $- \frac{e^{x} \sin{\left(x \right)}}{2} - \frac{e^{x} \cos{\left(x \right)}}{2}$

   The result is: $- e^{x} \cos{\left(x \right)}$

3. Add the constant of integration:

   $$- e^{x} \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- e^{x} \cos{\left(x \right)}+ \mathrm{constant}$$