To solve the integral \(\int \frac{3\sin{x} + 4\cos{x}}{5\sin{x} + 6\cos{x}} \, dx\), we can use a substitution method. Let's go through the steps.

### Step 1: Substitution
Let:
\[
u = 5\sin{x} + 6\cos{x}
\]
Then, differentiate \(u\) with respect to \(x\):
\[
\frac{du}{dx} = 5\cos{x} - 6\sin{x}
\]
So,
\[
du = (5\cos{x} - 6\sin{x}) \, dx
\]

### Step 2: Rewrite the Integral
Now, express the numerator \(3\sin{x} + 4\cos{x}\) in a form related to \(\frac{du}{dx}\). Notice that:
\[
3\sin{x} + 4\cos{x} = A(5\cos{x} - 6\sin{x}) + B(5\sin{x} + 6\cos{x})
\]
We need to find constants \(A\) and \(B\) such that:
\[
3\sin{x} + 4\cos{x} = A(5\cos{x} - 6\sin{x}) + B(5\sin{x} + 6\cos{x})
\]
Expanding the right-hand side:
\[
A(5\cos{x} - 6\sin{x}) + B(5\sin{x} + 6\cos{x}) = (5A + 6B)\cos{x} + (-6A + 5B)\sin{x}
\]

### Step 3: Equating Coefficients
For the equation to hold for all \(x\), the coefficients of \(\sin{x}\) and \(\cos{x}\) on both sides must be equal. Thus:
\[
-6A + 5B = 3 \quad \text{(coefficient of } \sin{x}\text{)}
\]
\[
5A + 6B = 4 \quad \text{(coefficient of } \cos{x}\text{)}
\]

### Step 4: Solve for \(A\) and \(B\)


Learn how to integrate (3sinx + 4cosx)/(5sinx + 6cosx) using trigonometric substitution. This detailed step-by-step solution covers advanced integration techniques. Suitable for advanced college-level mathematics.

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