Find: integral of e^(3x)sin(4x) dx

To find the integral $\int e^{3x} \sin(4x) \, dx$, we use the standard result for integrals of the form $\int e^{ax} \sin(bx) \, dx$.

Step 1: Identify the coefficients $a$ and $b$. The given integral is $\int e^{3x} \sin(4x) \, dx$. Comparing this to $\int e^{ax} \sin(bx) \, dx$, we have $a=3$ and $b=4$.

Step 2: Apply the standard integration formula. The standard formula for this type of integral is: $\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C$

Step 3: Substitute the values of $a$ and $b$ into the formula. Substitute $a=3$ and $b=4$: $\int e^{3x} \sin(4x) \, dx = \frac{e^{3x}}{3^2 + 4^2} (3 \sin(4x) - 4 \cos(4x)) + C$

Step 4: Simplify the expression. Calculate $3^2 + 4^2$: $3^2 + 4^2 = 9 + 16 = 25$ Substitute this back into the formula: $\int e^{3x} \sin(4x) \, dx = \frac{e^{3x}}{25} (3 \sin(4x) - 4 \cos(4x)) + C$

The final answer is $\boxed{\frac{e^{3x}}{25} (3 \sin(4x) - 4 \cos(4x)) + C}$.