To solve the integral $\int e^{5x} \sin(6x) \, dx$, we will use the method of integration by parts.

Step 1: Identify $u$ and $dv$

We'll set: $u = e^{5x}, \quad dv = \sin(6x) \, dx$

Now, find $du$ and $v$:

1. Differentiate $u$: $du = 5e^{5x} \, dx$

2. Integrate $dv$ to find $v$: $v = \int \sin(6x) \, dx = -\frac{1}{6} \cos(6x)$

Step 2: Apply the Integration by Parts Formula

The integration by parts formula is: $\int u \, dv = uv - \int v \, du$

Substitute the values: $\int e^{5x} \sin(6x) \, dx = e^{5x} \left(-\frac{1}{6} \cos(6x)\right) - \int \left(-\frac{1}{6} \cos(6x)\right) \cdot 5e^{5x} \, dx$

Simplify the expression: $\int e^{5x} \sin(6x) \, dx = -\frac{1}{6} e^{5x} \cos(6x) + \frac{5}{6} \int e^{5x} \cos(6x) \, dx$

Step 3: Solve the Remaining Integral

Now, we have to solve: $\int e^{5x} \cos(6x) \, dx$ We use integration by parts again. Set: $u = e^{5x}, \quad dv = \cos(6x) \, dx$

Then, find $du$ and $v$:

1. $du = 5e^{5x} \, dx$
2. $v = \int \cos(6x) \, dx = \frac{1}{6} \sin(6x)$

Apply integration by parts again: $\int e^{5x} \cos(6x) \, dx = e^{5x} \left(\frac{1}{6} \sin(6x)\right) - \int \frac{1}{6} \sin(6x) \cdot 5e^{5x} \, dx$ Simplify: $\int e^{5x} \cos(6x) \, dx = \frac{1}{6} e^{5x} \sin(6x) - \frac{5}{6} \int e^{5x} \sin(6x) \, dx$

Step 4: Combine and Solve

Substitute this back into the original equation: $\int e^{5x} \sin(6x) \, dx = -\frac{1}{6} e^{5x} \cos(6x) + \frac{5}{6} \left(\frac{1}{6} e^{5x} \sin(6x) - \frac{5}{6} \int e^{5x} \sin(6x) \, dx\right)$

Now, group the integrals involving $\int e^{5x} \sin(6x) \, dx$ on one side: $\int e^{5x} \sin(6x) \, dx \left(1 + \frac{25}{36}\right) = -\frac{1}{6} e^{5x} \cos(6x) + \frac{5}{36} e^{5x} \sin(6x)$

Simplify: $\int e^{5x} \sin(6x) \, dx \left(\frac{61}{36}\right) = -\frac{1}{6} e^{5x} \cos(6x) + \frac{5}{36} e^{5x} \sin(6x)$

Finally, solve for the integral: $\int e^{5x} \sin(6x) \, dx = \frac{-6e^{5x} \cos(6x) + 5e^{5x} \sin(6x)}{61} + C$ where $C$ is the constant of integration.

Final Answer

$\int e^{5x} \sin(6x) \, dx = \frac{1}{61} e^{5x} \left(-6 \cos(6x) + 5 \sin(6x)\right) + C$

Would you like further details on any of the steps or additional examples?

Related Questions

1. How would the approach change if the integrand was $e^{ax} \sin(bx)$?
2. What if the integrand involved a cosine instead of a sine, like $e^{5x} \cos(6x)$?
3. How does integration by parts work for trigonometric integrals in general?
4. Can this method be applied to integrals involving hyperbolic functions?
5. How does the reduction formula for $\int e^{ax} \sin(bx) \, dx$ simplify this problem?

Tip

When facing integrals of products like $e^{ax} \sin(bx)$, expect to use integration by parts twice, and always look for patterns that simplify the algebraic manipulation.