Integral of 5e^(-x)cos(2x) dx

1. Let $u = - x$.

   Then let $du = - dx$ and substitute $- 5 du$:

   $$\int \left(- 5 e^{u} \cos{\left(2 u \right)}\right)\, du$$

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

      $$\int e^{u} \cos{\left(2 u \right)}\, du = - 5 \int e^{u} \cos{\left(2 u \right)}\, du$$

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

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

            Let $u{\left(u \right)} = \cos{\left(2 u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

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

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

            Let $u{\left(u \right)} = - 2 \sin{\left(2 u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

            Then $\int e^{u} \cos{\left(2 u \right)}\, du = 2 e^{u} \sin{\left(2 u \right)} + e^{u} \cos{\left(2 u \right)} + \int \left(- 4 e^{u} \cos{\left(2 u \right)}\right)\, du$.

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

            $$5 \int e^{u} \cos{\left(2 u \right)}\, du = 2 e^{u} \sin{\left(2 u \right)} + e^{u} \cos{\left(2 u \right)}$$

            Therefore,

            $$\int e^{u} \cos{\left(2 u \right)}\, du = \frac{2 e^{u} \sin{\left(2 u \right)}}{5} + \frac{e^{u} \cos{\left(2 u \right)}}{5}$$

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

   Now substitute $u$ back in:

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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