Question

use a table of integrals to evaluate the integral. $int e^{5x}sin(3x)dx$ + c

Explanation:

Step1: Recall integral formula

The formula for $\int e^{ax}\sin(bx)dx=\frac{e^{ax}}{a^{2}+b^{2}}(a\sin(bx)-b\cos(bx))+C$.
Here $a = 5$ and $b=3$.

Step2: Substitute values

Substitute $a = 5$ and $b = 3$ into the formula.
We get $\frac{e^{5x}}{5^{2}+3^{2}}(5\sin(3x)-3\cos(3x))+C$.

Step3: Simplify

Calculate $5^{2}+3^{2}=25 + 9=34$.
So the integral is $\frac{e^{5x}}{34}(5\sin(3x)-3\cos(3x))+C$.

Answer:

$\frac{e^{5x}}{34}(5\sin(3x)-3\cos(3x))+C$