To evaluate the integral

$\int e^{4x} \sin(6x) \, dx,$

we'll need to use the technique of integration by parts or apply the tabular method for repetitive integration by parts. Alternatively, we can utilize the method of solving differential equations by letting the integral equal an unknown function $I(x)$ and differentiating it to find a solution.

Here's how we can approach it:

Step 1: Set up the integral

Let: $I = \int e^{4x} \sin(6x) \, dx.$

Step 2: Use Integration by Parts

To proceed, we will apply integration by parts twice. However, this can be simplified using the formula:

$\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C.$

For this problem:

• $a = 4$
• $b = 6$

So we substitute these values:

$I = \frac{e^{4x}}{4^2 + 6^2} (4 \sin(6x) - 6 \cos(6x)).$

Step 3: Simplify the Result

Calculate $4^2 + 6^2$: $4^2 + 6^2 = 16 + 36 = 52.$

Thus,

$I = \frac{e^{4x}}{52} (4 \sin(6x) - 6 \cos(6x)) + C.$

Final Answer

$\int e^{4x} \sin(6x) \, dx = \frac{e^{4x}}{52} (4 \sin(6x) - 6 \cos(6x)) + C.$

Would you like more details or further explanation on any step?

Related Questions for Further Practice:

1. How would you evaluate $\int e^{5x} \cos(7x) \, dx$?
2. What is the general approach for integrating functions of the form $e^{ax} \sin(bx)$?
3. How does integration by parts work for trigonometric integrals involving exponential functions?
4. Can you derive a general formula for $\int e^{ax} \cos(bx) \, dx$?
5. What is the significance of using the method of differential equations in integrals involving products of exponential and trigonometric functions?

Tip: For integrals involving products of exponential and trigonometric functions, it can be efficient to use the formula derived from integration by parts applied twice, as it saves time.