Question

Find: \[ \int \frac{\cos x}{(2+\sin x)(4+\sin x)}\,dx \] 

Hint:

Look for a derivative pair (like \(\sin x\) and \(\cos x\)) to simplify the integral into a rational form. Partial fraction decomposition is the standard tool for integrating rational functions with factorable denominators.

Answer 1

Concept:
• This is an integration problem solvable by substitution followed by partial fraction decomposition.
• The presence of \(\cos x\) in the numerator suggests substituting \(t = \sin x\).

Step 1: Apply substitution
Let \(t = \sin x\). Then, \(dt = \cos x \, dx\). The integral becomes: \[ I = \int \frac{1}{(2 + t)(4 + t)} \, dt \]

Step 2: Apply partial fractions
Let \(\frac{1}{(2 + t)(4 + t)} = \frac{A}{t + 2} + \frac{B}{t + 4}\). Multiplying through: \[ 1 = A(t + 4) + B(t + 2) \] To find \(A\), set \(t = -2\): \(1 = 2A \implies A = 1/2\). To find \(B\), set \(t = -4\): \(1 = -2B \implies B = -1/2\). So: \[ \frac{1}{(t + 2)(t + 4)} = \frac{1}{2(t + 2)} - \frac{1}{2(t + 4)} \]

Step 3: Integrate
\[ I = \frac{1}{2} \int \frac{1}{t + 2} \, dt - \frac{1}{2} \int \frac{1}{t + 4} \, dt \] \[ I = \frac{1}{2} \log|t + 2| - \frac{1}{2} \log|t + 4| + C \] Using log properties: \[ I = \frac{1}{2} \log \left| \frac{t + 2}{t + 4} \right| + C \]

Step 4: Back-substitute for \(x\)
\[ I = \frac{1}{2} \log \left| \frac{\sin x + 2}{\sin x + 4} \right| + C \]