evaluate ∫ sin(3𝑥)sin(4𝑥) 𝑑𝑥 using Euler's formula and in terms of e and i

To evaluate the integral $\int \sin(3x)\sin(4x) \, dx$ using Euler's formula and in terms of $e$ and $i$, we can follow these steps:

1. **Express the sines in terms of exponential functions using Euler's formula:**

\[ \sin(a) = \frac{e^{ia} - e^{-ia}}{2i} \]

Applying this to $\sin(3x)$ and $\sin(4x)$:

\[ \sin(3x) = \frac{e^{i3x} - e^{-i3x}}{2i} \]
\[ \sin(4x) = \frac{e^{i4x} - e^{-i4x}}{2i} \]

2. **Multiply the two expressions:**

\[ \sin(3x)\sin(4x) = \left( \frac{e^{i3x} - e^{-i3x}}{2i} \right) \left( \frac{e^{i4x} - e^{-i4x}}{2i} \right) \]

3. **Simplify the product:**

\[ \sin(3x)\sin(4x) = \frac{1}{(2i)^2} (e^{i3x} - e^{-i3x})(e^{i4x} - e^{-i4x}) \]
\[ \sin(3x)\sin(4x) = \frac{1}{-4} (e^{i3x}e^{i4x} - e^{i3x}e^{-i4x} - e^{-i3x}e^{i4x} + e^{-i3x}e^{-i4x}) \]
\[ \sin(3x)\sin(4x) = -\frac{1}{4} (e^{i7x} - e^{-ix} - e^{ix} + e^{-i7x}) \]

4. **Separate the exponentials:**

\[ \sin(3x)\sin(4x) = -\frac{1}{4} (e^{i7x} + e^{-i7x} - e^{ix} - e^{-ix}) \]

5. **Rewrite in terms of cosines:**

\[ e^{i7x} + e^{-i7x} = 2\cos(7x) \]
\[ e^{ix} + e^{-ix} = 2\cos(x) \]

Therefore:

\[ \sin(3x)\sin(4x) = -\frac{1}{4} (2\cos(7x) - 2\cos(x)) \]
\[ \sin(3x)\sin(4x) = -\frac{1}{4} \cdot 2 (\cos(7x) - \cos(x)) \]
\[ \sin(3x)\sin(4x) = -\frac{1}{2} (\cos(7x) - \cos(x)) \]

6. **Integrate each term separately:**

\[ \int \sin(3x)\sin(4x) \, dx = -\frac{1}{2} \int (\cos(7x) - \cos(x)) \, dx \]

\[ \int \cos(7x) \, dx = \frac{\sin(7x)}{7} \]
\[ \int \cos(x) \, dx = \sin(x) \]

So:

\[ \int \sin(3x)\sin(4x) \, dx = -\frac{1}{2} \left( \frac{\sin(7x)}{7} - \sin(x) \right) + C \]

Simplifying the constant $C$:

\[ \int \sin(3x)\sin(4x) \, dx = -\frac{1}{14} \sin(7x) + \frac{1}{2} \sin(x) + C \]

Thus, the integral is:

\[ \boxed{\int \sin(3x)\sin(4x) \, dx = -\frac{1}{14} \sin(7x) + \frac{1}{2} \sin(x) + C} \]

Would you like more details or have any questions about this process? Here are some related questions you might want to explore:

1. How can you use trigonometric identities to solve integrals involving products of sine and cosine?
2. What are the steps for integrating functions using partial fractions?
3. How can you derive Euler's formula from the Taylor series?
4. What are the methods for evaluating integrals using substitution?
5. How do you integrate functions involving hyperbolic sine and cosine?

**Tip:** When integrating products of trigonometric functions, converting them to exponential form using Euler's formula can simplify the process, especially for complex integrals.

Learn how to evaluate the integral $\int \sin(3x)\sin(4x) \, dx$ using Euler's formula and in terms of $e$ and $i$. Follow a step-by-step solution involving trigonometric identities and integration techniques. Suitable for advanced high school students and beyond.

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