Question

The indefinite integral of (12 sin x + 5 cos x)^–1 is, for any arbitrary constant –

• A. 1/13 log $\left| \tan\left( \frac{x}{2} + \frac{1}{2}\tan^{- 1}\frac{5}{12} \right) \right|$ + c
• B. 1/13 log $\left| \tan\left( \frac{x}{2} + \frac{1}{2}\tan^{- 1}\frac{12}{5} \right) \right|$ + c
• C. 1/13 log $\left| \tan\left( x + \tan^{- 1}\frac{5}{12} \right) \right|$ + c
• D. 1/13 log $\left| \tan\left( x + \tan^{- 1}\frac{12}{5} \right) \right|$ + c

Answer 1

Answer: (A)

1/13 log $\left| \tan\left( \frac{x}{2} + \frac{1}{2}\tan^{- 1}\frac{5}{12} \right) \right|$ + c

Answer 2

With a = tan^–1$\frac{5}{12}$, sin a = $\frac{5}{13}$ and cos a = $\frac{12}{13}$

= $\int_{}^{}\frac{dx}{13\sin(x + \alpha)}$

= $\frac { 1 } { 13 } \int \frac { \frac { 1 } { 2 } \sec ^ { 2 } \frac { x + \alpha } { 2 } } { \tan \frac { x + \alpha } { 2 } } d x$

= $\frac{1}{13}$ log $\left| \tan\left( \frac{x}{2} + \frac{1}{2}\tan^{- 1}\frac{5}{12} \right) \right|$ + c