$$ \int_{}^{}e^{5x}\sin\left(6x\right)\,\mathrm{d}x $$

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.

Explore the step-by-step solution for the integral ∫ e^(5x) * sin(6x) dx using integration by parts. Understand how to apply this method to solve complex trigonometric integrals involving exponential functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation