Question

integration of dx/√25-16x² ?

Answer 1

int dx/(√25-16x²)

int dx/(√5²-(4x)²)

arcsin(4x/5)/4+C

Answer 2

x/√(25 - 16x²) 
= ¼ ∫dx/√(5²/4² - x²) 

Let x = 5sin(u)/4 ... hence u = sinֿ¹(4x/5) 
dx = 5cos(u)du/4 

= ¼ ∫(5cos(u)du/4)/√(5²/4² - 5²sin²(u)/4²) 
= ¼∫(5cos(u)du/4) / (5/4)√(1 - sin²(u)) 
= ¼ ∫cos(u)du / √(cos²(u)) 
= ¼ ∫cos(u)du / cos(u) 
= ¼ ∫du 
= ¼u + C 
= ¼ sinֿ¹(4x/5) + C