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

Answer

To evaluate the integral

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

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral for clarity:

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

### Step 2: Split the Integral
We can split the integral into two parts:

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

### Step 3: Integrate the First Part
The first integral is straightforward:

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

### Step 4: Integrate the Second Part
Now, we need to integrate the second part:

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

### Step 5: Solve Each Integral
Now we solve each integral separately:

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

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

### Step 6: Combine the Results
Putting these results back into the expression, we have:

\[ \frac{5}{13} \left( -3 \cos x - 2 \sin x \right) = -\frac{15}{13} \cos x - \frac{10}{13} \sin x. \]

### Step 7: Combine All Parts
Now, combine all parts of the integral:

\[ \int \left( \frac{12}{13} - \frac{5}{13}(-3 \sin x + 2 \cos x) \right) dx = \frac{12}{13} x - \frac{15}{13} \cos x - \frac{10}{13} \sin x + C, \]

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

### Final Answer
Thus, the final answer is:

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

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