Integral of xe^xsinxdx dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

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

   Then $\operatorname{du}{\left(x \right)} = 1$.

   To find $v{\left(x \right)}$:

   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,

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

   Now evaluate the sub-integral.

2. Integrate term-by-term:

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

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

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

         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,

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

      So, the result is: $\frac{e^{x} \sin{\left(x \right)}}{4} - \frac{e^{x} \cos{\left(x \right)}}{4}$

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

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

      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: $- \frac{e^{x} \sin{\left(x \right)}}{4} - \frac{e^{x} \cos{\left(x \right)}}{4}$

   The result is: $- \frac{e^{x} \cos{\left(x \right)}}{2}$

3. Now simplify:

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

4. Add the constant of integration:

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

The answer is: