Integral of -sin(x)*6*e^(x) dx

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

   $$\int - \sin{\left(x \right)} 6 e^{x}\, dx = 6 \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx$$

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

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

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

   So, the result is: $- 3 e^{x} \sin{\left(x \right)} + 3 e^{x} \cos{\left(x \right)}$

2. Now simplify:

   $$3 \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}$$

3. Add the constant of integration:

   $$3 \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$$

The answer is:

$$3 \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$$