Similar to last problem

Question

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "DNE"

∫ with 4 on top and -oo on bottom with (1)/(x^2 +1) dx

Answer 1

∫_{(-∞ to 4)} [1 / (x²+1)]dx = lim_b→-∞ ∫_{(b to 4)} [1 / (x²+1)]dx = lim_b→-∞ [Arctanx]_{(b to 4)}

= lim_b→-∞ [Arctan4 - Arctanb] = Arctan4 - (-π/2) = Arctan4 + π/2

Answer 2

hi,

∫ with 4 on top and -oo on bottom with (1)/(x^2 +1) dx

I= ∫(1)/(x^2 +1) dx ====> as you know x= tanθ dx = sec^2 θ dθ then replace these info to the origin question .

I= ∫(1)/(x^2 +1) dx = ∫ sec^θ 2 dθ /( 1 + tanθ ^2) as you know 1 + tanθ ^2 = sec^2 θ after simplify your integral will be

∫dθ = θ then you know x must be changed to the x SO x= tanθ ====> θ = arc tanx replace -∞ and 4

I= arc tan4 - arc tan-∞ ======>

I= arc tan4 + π/2

your integral is convergent and the answer of integral is I = arc tan4 + π/2

I hope it is useful.

Minoo