To solve the integral, we use the substitution method followed by partial fractions.
Step 1: Substitution
Let u=cosθ. Then, the derivative is du=−sinθdθ, which means sinθdθ=−du.
Substituting this into the integral, we get:
∫(2+u)(3+4u)−du=−∫(2+u)(3+4u)du
Step 2: Partial Fraction Decomposition
We decompose the integrand (2+u)(3+4u)1 into partial fractions:
(2+u)(3+4u)1=5(3+4u)4−5(2+u)1
Step 3: Integration
Now substitute this back into the integral:
−∫(5(3+4u)4−5(2+u)1)du
=−(54⋅41ln∣3+4u∣−51ln∣2+u∣)
=−(51ln∣3+4u∣−51ln∣2+u∣)
=51ln∣2+u∣−51ln∣3+4u∣
Step 4: Back-substitution and Comparison
Substituting u=cosθ back, we have:
51ln∣2+cosθ∣−51ln∣3+4cosθ∣
Comparing this with the given form Aln∣2+cosθ∣+Bln∣3+4cosθ∣, we find:
A=51
Therefore, the correct option is (c).
