Evaluate the following Integrals.
`int(12)/(13) - (5)/(13) (-3 sin x + 2 cos x)dx`

Answer

To evaluate the integral

\[ I = \int \left( \frac{12}{13} - \frac{5}{13}(-3 \sin x + 2 \cos x) \right) dx, \]

we can break it down into simpler parts.

### Step 1: Split the Integral
We can rewrite the integral by distributing the terms:

\[ I = \int \frac{12}{13} \, dx - \frac{5}{13} \int (-3 \sin x + 2 \cos x) \, dx. \]

### Step 2: Integrate the Constant Term
The integral of a constant \( \frac{12}{13} \) is:

\[ \int \frac{12}{13} \, dx = \frac{12}{13} x. \]

### Step 3: Integrate the Sine and Cosine Terms
Next, we need to integrate the sine and cosine terms. We can split this into two separate integrals:

\[ I = \frac{12}{13} x - \frac{5}{13} \left( \int -3 \sin x \, dx + \int 2 \cos x \, dx \right). \]

Now, we compute each integral:

1. For \( \int -3 \sin x \, dx \):
\[ \int -3 \sin x \, dx = -3 \left( -\cos x \right) = 3 \cos x. \]

2. For \( \int 2 \cos x \, dx \):
\[ \int 2 \cos x \, dx = 2 \sin x. \]

### Step 4: Combine the Results
Now we can substitute these results back into our expression for \( I \):

\[ I = \frac{12}{13} x - \frac{5}{13} \left( 3 \cos x + 2 \sin x \right). \]

### Step 5: Distribute the Coefficient
Distributing \( -\frac{5}{13} \):

\[ I = \frac{12}{13} x - \frac{5}{13} \cdot 3 \cos x - \frac{5}{13} \cdot 2 \sin x. \]

This simplifies to:

\[ I = \frac{12}{13} x - \frac{15}{13} \cos x - \frac{10}{13} \sin x + C, \]

where \( C \) is the constant of integration.

### Final Answer
Thus, the final result for the integral is:

\[ I = \frac{12}{13} x - \frac{15}{13} \cos x - \frac{10}{13} \sin x + C. \]

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