Integral of (e^(5x))cos(7x) dx

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

   1. For the integrand $e^{5 x} \cos{\left(7 x \right)}$:

      Let $u{\left(x \right)} = \cos{\left(7 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$.

      Then $\int e^{5 x} \cos{\left(7 x \right)}\, dx = \frac{e^{5 x} \cos{\left(7 x \right)}}{5} - \int \left(- \frac{7 e^{5 x} \sin{\left(7 x \right)}}{5}\right)\, dx$.

   2. For the integrand $- \frac{7 e^{5 x} \sin{\left(7 x \right)}}{5}$:

      Let $u{\left(x \right)} = - \frac{7 \sin{\left(7 x \right)}}{5}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$.

      Then $\int e^{5 x} \cos{\left(7 x \right)}\, dx = \frac{7 e^{5 x} \sin{\left(7 x \right)}}{25} + \frac{e^{5 x} \cos{\left(7 x \right)}}{5} + \int \left(- \frac{49 e^{5 x} \cos{\left(7 x \right)}}{25}\right)\, dx$.

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

      $$\frac{74 \int e^{5 x} \cos{\left(7 x \right)}\, dx}{25} = \frac{7 e^{5 x} \sin{\left(7 x \right)}}{25} + \frac{e^{5 x} \cos{\left(7 x \right)}}{5}$$

      Therefore,

      $$\int e^{5 x} \cos{\left(7 x \right)}\, dx = \frac{7 e^{5 x} \sin{\left(7 x \right)}}{74} + \frac{5 e^{5 x} \cos{\left(7 x \right)}}{74}$$

2. Now simplify:

   $$\frac{\left(7 \sin{\left(7 x \right)} + 5 \cos{\left(7 x \right)}\right) e^{5 x}}{74}$$

3. Add the constant of integration:

   $$\frac{\left(7 \sin{\left(7 x \right)} + 5 \cos{\left(7 x \right)}\right) e^{5 x}}{74}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(7 \sin{\left(7 x \right)} + 5 \cos{\left(7 x \right)}\right) e^{5 x}}{74}+ \mathrm{constant}$$