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Answer to Question #19914 in Calculus for sakura

Question #19914
antiderivative of sqrt(5 + 4x - x^2)

using chain rule or integration by parts.
Can i solve it not using trigonometry?
Expert's answer
Possible intermediate steps:
integralsqrt(-x^2+4 x+5) dx
For the integrand sqrt(-x^2+4 x+5), complete the square:
= integral sqrt(9-(x-2)^2) dx
For the integrand sqrt(9-(x-2)^2), substitute u = x-2and du = dx:
= integral sqrt(9-u^2) du
For the integrand sqrt(9-u^2), substitute u = 3 sin(s)and du = 3 cos(s) ds. Then sqrt(9-u^2) = sqrt(9-9 sin^2(s)) = 3cos(s) and s = sin^(-1)(u/3):
= 9 integralcos^2(s) ds
Write cos^2(s) as 1/2 cos(2 s)+1/2:
= 9 integral (1/2cos(2 s)+1/2) ds
Integrate the sum term by term and factor out constants:
= 9 integral 1/2ds+9/2 integral cos(2 s) ds For theintegrand cos(2 s), substitute p = 2 s and dp = 2 ds:
= 9/4 integral cos(p) dp+9 integral 1/2 ds Theintegral of cos(p) is sin(p):
= (9 sin(p))/4+9integral 1/2 ds
The integral of 1/2 is s/2:
= (9 sin(p))/4+(9s)/2+constant
Substitute back for p = 2 s:
= (9 s)/2+9/4sin(2 s)+constant
Substitute back for s = sin^(-1)(u/3):
= 1/2 usqrt(9-u^2)+9/2 sin^(-1)(u/3)+constant Substitute back for u = x-2:
= 1/2 (x-2)sqrt(-x^2+4 x+5)+9/2 sin^(-1)((x-2)/3)+constant Factor the answer a different
way:
Answer: |
| = 1/2 ((x-2) sqrt(-x^2+4 x+5)+9sin^(-1)((x-2)/3))+constant

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