Pls help me with some calculus problems? Greatly appreciated!?

Question

Hi, i was wondering if any of you math-tastic people out there could help me figure out these math problems.

they are part of the chapter called "trig substitution"

1) integral of (dx/[(25-x^2)^(3/2)])
where x=5 sin (thetha)

2) integral of ((25-x^2)^(1/2))/x dx
where x= 5 sin (thetha)

thank you!!

Answer 1

1)
If x = 5 sinθ, dx = 5 cosθ dθ

then (25 - x²)^(3/2) = (25 - 25 sin²θ)^(3/2)
= (25cos²θ)^(3/2)
= 125 cos³θ

and dx / (25 - x²)^(3/2) = dθ / 25cos²θ

the integral is then tanθ / 25

2)
[(25 - x²)^(1/2) / x] dx = [5 cosθ / 5 sinθ] * 5 cosθ dθ
= 5 cos²θ / sinθ dθ

and ... what is the integral ??

the answer is (difficult) : 5 [log (tan(0.5θ) + cosθ]