Integral of exp(7x)*sin(x) dx

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

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

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

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

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

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

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

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

      $$\frac{50 \int e^{7 x} \sin{\left(x \right)}\, dx}{49} = \frac{e^{7 x} \sin{\left(x \right)}}{7} - \frac{e^{7 x} \cos{\left(x \right)}}{49}$$

      Therefore,

      $$\int e^{7 x} \sin{\left(x \right)}\, dx = \frac{7 e^{7 x} \sin{\left(x \right)}}{50} - \frac{e^{7 x} \cos{\left(x \right)}}{50}$$

2. Now simplify:

   $$\frac{\left(7 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{7 x}}{50}$$

3. Add the constant of integration:

   $$\frac{\left(7 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{7 x}}{50}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(7 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{7 x}}{50}+ \mathrm{constant}$$