Question

Oblicz całkę nieoznaczoną, stosując odpowiednie podstawienie

∫(dx) / ( 4 (x^2) - 20x +34 )

Answer 1

∫(dx/4x² -20x +34)

Δ = 400 -4*4*34 = 400 - 544 = - 144

p = 20/8 = 5/2

q = 144/8 = 9

zatem

4x² - 20x + 34 = 4*[(x +5/2)² +9]

czyli

I = ∫(dx/4x² - 20x +34) = ∫[dx/(4*[(x +5/2)² +9] =

=(1/4)∫ (dx/[(x + 5/2)² + 9/4])

Podstawienie

x + 5/2 = (3/2)t  ---->  t = (2/3)x +5/3

dx = (3/2) dt

zatem I = (1/4)∫(3/2)dt/( (9/4)t² +9/4] =

= (1/4)*(3/2)*(4/9) ∫ dt/(t² + 1) = (1/6)∫ dt/(t² +1) =

= (1/6) *arctg t + c' = (1/6) *arctg[(2/3)x +(5/3)] + C

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