To find the value of B, we proceed with the integration using substitution and partial fractions.
Step 1: Substitution
Let u=cosθ. Then, the differential is du=−sinθdθ, which implies sinθdθ=−du. Substituting these into the integral gives:
∫(2+u)(3+4u)−du=−∫(2+u)(3+4u)du
Step 2: Partial Fraction Decomposition
We express the integrand as a sum of partial fractions:
(2+u)(3+4u)1=5(3+4u)4−5(2+u)1
Step 3: Integration
Now, integrate the decomposed expression:
−∫(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 into the result:
51ln∣2+cosθ∣−51ln∣3+4cosθ∣
Comparing this with the given form Aln∣2+cosθ∣+Bln∣3+4cosθ∣, we identify the coefficients:
A=51,B=−51
Therefore, the value of B is −51, and the correct option is (b).
