Question

∫[dx/{(x + 2)^(12/13)(x – 5)^(14/13)}] = _________ + c 

(a) [(– 13)/7][(x + 2)/(x – 5)]^(1/13) 

(b) [(13)/7][(x + 2)/(x – 5)]^(1/13) 

(c) [(13)/7][(x – 5)/(x + 2)]^(1/13) 

(d) [(– 13)/7][(x – 5)/(x – 2)]^(1/13)

Answer 1

The correct option (a) [(– 13)/7][(x + 2)/(x – 5)]^(1/13)

Explanation:

[1/{(x + 2)^(12/13)(x – 5)^(14/13)}] = [1/({(x + 2)/(x – 5)}^(12/13) ∙ (x – 5)²)] 

Let [(x + 2)/(x – 5)] = t 

∴ [{(x – 5)(1) – (x + 2)}/(x – 5)²]dx = dt 

∴ [(– 7 dx)/(x – 5)²] = dt 

∴ I = ∫[dt/t^(12/13)] [– (1/7)] 

= – (1/7)∫t–(12/13)dt 

= – (1/7) [t^(1/13)/(1/13)] + c 

= – (13/7)[(x + 2)/(x – 5)]^(1/13) + c