Answer
a) False b) True c) True
Work Step by Step
Differentiate each option to check the original integral:
a) We have: $\dfrac{d}{d\theta}[\dfrac{\sec ^3 \theta}{3} +C]= \sec^2 \theta \sec \theta \tan \theta \ne \tan \theta \sec^2 \theta$
b) We have: $\dfrac{d}{d\theta}[\dfrac{1}{2} \tan^2 \theta +C]= \dfrac{1}{2} (2 \sec^2 \theta \tan \theta) = \tan \theta \sec^2 \theta$
c) We have: $\dfrac{d}{d\theta}[\dfrac{1}{2} \sec^2 \theta +C]= \dfrac{1}{2} (2 \sec \theta \sec \theta \tan \theta) = \tan \theta \sec^2 \theta$
Hence, a) False, b) True, c) True
