Question

\( \int \frac{13\cos 2x - 9\sin 2x}{3\cos 2x - 4\sin 2x} dx = \)

• A. \( 3x - \frac{1}{2}\log|3\cos 2x - 4\sin 2x| + c \)
• B. \( \frac{x}{2} - 3\log|3\cos 2x - 4\sin 2x| + c \)
• C. \( 3x + \frac{1}{2}\log|3\cos 2x - 4\sin 2x| + c \)
• D. \( x + \frac{3}{2}\log|3\cos 2x - 4\sin 2x| + c \)

Hint:

For integrals of the form \( \int \frac{A f(x) + B f'(x)}{C f(x) + D f'(x)} dx \) or more specifically \( \int \frac{a\cos kx + b\sin kx}{p\cos kx + q\sin kx} dx \), assume: Numerator \( = L \cdot (\text{Denominator}) + M \cdot (\text{Derivative of Denominator}) \). Solve for \(L\) and \(M\) by comparing coefficients. The integral then becomes \( \int (L + M \frac{\text{Derivative}}{\text{Denominator}}) dx = Lx + M \log|\text{Denominator}| + C \).

Answer 1

The Correct Option is A

This integral is of the form \( \int \frac{A\cos kx + B\sin kx}{C\cos kx + D\sin kx} dx \).
Let Numerator = \( L \cdot (\text{Denominator}) + M \cdot (\text{Derivative of Denominator}) \).
Let \( N(x) = 13\cos 2x - 9\sin 2x \).
Let \( D(x) = 3\cos 2x - 4\sin 2x \).
Derivative of Denominator: \( D'(x) = \frac{d}{dx}(3\cos 2x - 4\sin 2x) \) \[ = 3(-\sin 2x \cdot 2) - 4(\cos 2x \cdot 2) = -6\sin 2x - 8\cos 2x \] So, we write \( 13\cos 2x - 9\sin 2x = L(3\cos 2x - 4\sin 2x) + M(-8\cos 2x - 6\sin 2x) \).
Equate coefficients of \( \cos 2x \) and \( \sin 2x \): Coeff of \( \cos 2x \): \( 13 = 3L - 8M \cdots (1) \) Coeff of \( \sin 2x \): \( -9 = -4L - 6M \implies 9 = 4L + 6M \cdots (2) \) Solve for L and M.
Multiply (1) by 6 and (2) by 8: \( 78 = 18L - 48M \) \( 72 = 32L + 48M \) Add these two equations: \( 78+72 = 18L+32L \implies 150 = 50L \implies L=3 \).
Substitute \(L=3\) into (1): \( 13 = 3(3) - 8M \implies 13 = 9 - 8M \implies 4 = -8M \implies M = -4/8 = -1/2 \).
So the integral becomes: \[ \int \frac{L \cdot D(x) + M \cdot D'(x)}{D(x)} dx = \int \left( L + M \frac{D'(x)}{D(x)} \right) dx \] \[ = \int L dx + M \int \frac{D'(x)}{D(x)} dx = Lx + M \log|D(x)| + c \] Substitute \( L=3 \) and \( M=-1/2 \): \[ = 3x - \frac{1}{2} \log|3\cos 2x - 4\sin 2x| + c \] This matches option (1).