Integral of e^(3*x)*sin(2*x) dx

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

   $$\int 2 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}\, dx$$

   1. Use integration by parts:

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

      Let $u{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{3 x}$.

      Then $\operatorname{du}{\left(x \right)} = - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$.

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

      1. Let $u = 3 x$.

         Then let $du = 3 dx$ and substitute $\frac{du}{3}$:

         $$\int \frac{e^{u}}{3}\, du$$

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

            $$\text{False}$$

            1. The integral of the exponential function is itself.

               $$\int e^{u}\, du = e^{u}$$

            So, the result is: $\frac{e^{u}}{3}$

         Now substitute $u$ back in:

         $$\frac{e^{3 x}}{3}$$

      Now evaluate the sub-integral.

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

      $$\int \frac{\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{3 x}}{3}\, dx = \frac{\int \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{3 x}\, dx}{3}$$

      1. Rewrite the integrand:

         $$\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{3 x} = - e^{3 x} \sin^{2}{\left(x \right)} + e^{3 x} \cos^{2}{\left(x \right)}$$

      2. Integrate term-by-term:

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

            $$\int \left(- e^{3 x} \sin^{2}{\left(x \right)}\right)\, dx = - \int e^{3 x} \sin^{2}{\left(x \right)}\, dx$$

            1. Don't know the steps in finding this integral.

               But the integral is

               $$\frac{11 e^{3 x} \sin^{2}{\left(x \right)}}{39} - \frac{2 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}}{13} + \frac{2 e^{3 x} \cos^{2}{\left(x \right)}}{39}$$

            So, the result is: $- \frac{11 e^{3 x} \sin^{2}{\left(x \right)}}{39} + \frac{2 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}}{13} - \frac{2 e^{3 x} \cos^{2}{\left(x \right)}}{39}$

         1. Don't know the steps in finding this integral.

            But the integral is

            $$\frac{2 e^{3 x} \sin^{2}{\left(x \right)}}{39} + \frac{2 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}}{13} + \frac{11 e^{3 x} \cos^{2}{\left(x \right)}}{39}$$

         The result is: $- \frac{3 e^{3 x} \sin^{2}{\left(x \right)}}{13} + \frac{4 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}}{13} + \frac{3 e^{3 x} \cos^{2}{\left(x \right)}}{13}$

      So, the result is: $- \frac{e^{3 x} \sin^{2}{\left(x \right)}}{13} + \frac{4 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}}{39} + \frac{e^{3 x} \cos^{2}{\left(x \right)}}{13}$

   So, the result is: $\frac{2 e^{3 x} \sin^{2}{\left(x \right)}}{13} + \frac{6 e^{3 x} \sin{\left(x \right)} \cos{\left(x \right)}}{13} - \frac{2 e^{3 x} \cos^{2}{\left(x \right)}}{13}$

2. Now simplify:

   $$\frac{\left(3 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{3 x}}{13}$$

3. Add the constant of integration:

   $$\frac{\left(3 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{3 x}}{13}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(3 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{3 x}}{13}+ \mathrm{constant}$$