solve the integral of 19(sinx)(cosx)dx using integration by parts

To solve the integral \( \int 19 \sin(x) \cos(x) \, dx \) using integration by parts, we first need to apply a suitable strategy. However, this integral can be simplified before using integration by parts by recognizing the trigonometric identity:

\[
\sin(2x) = 2 \sin(x) \cos(x)
\]

So, we can rewrite the integral as:

\[
\int 19 \sin(x) \cos(x) \, dx = \int \frac{19}{2} \sin(2x) \, dx
\]

Now, let's solve this integral:

### Step 1: Simplify the Integral
\[
\int \frac{19}{2} \sin(2x) \, dx
\]

### Step 2: Integrate Using Substitution
Let \( u = 2x \), so \( du = 2dx \) or \( dx = \frac{du}{2} \).

Substituting into the integral, we get:

\[
\int \frac{19}{2} \sin(2x) \, dx = \frac{19}{2} \int \sin(u) \cdot \frac{du}{2} = \frac{19}{4} \int \sin(u) \, du
\]

### Step 3: Integrate
The integral of \( \sin(u) \) is \( -\cos(u) \), so:

\[
\frac{19}{4} \int \sin(u) \, du = -\frac{19}{4} \cos(u) + C
\]

### Step 4: Substitute Back \( u = 2x \)
Substituting \( u = 2x \) back into the equation, we get:

\[
-\frac{19}{4} \cos(2x) + C
\]

So, the integral \( \int 19 \sin(x) \cos(x) \, dx \) is:

\[
\boxed{-\frac{19}{4} \cos(2x) + C}
\]

Would you like a more detailed explanation or have any questions about the process? 

Here are five related questions you might consider:

1. How would you solve the integral of \( \sin(x) \cos(x) \) without using substitution?
2. What are some other trigonometric identities that can simplify integrals?
3. How does integration by parts work in cases where substitution is not applicable?
4. Can you derive the identity \( \sin(2x) = 2 \sin(x) \cos(x) \) from basic trigonometric principles?
5. What is the significance of the constant \( C \) in indefinite integrals?

**Tip:** Always check if a trigonometric identity can simplify the integral before choosing a method like integration by parts!

Learn how to solve the integral of 19sin(x)cos(x) using the trigonometric identity sin(2x) = 2sin(x)cos(x) and substitution. This step-by-step guide simplifies the process and is suitable for advanced high school or college-level calculus students.

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