2.7
1
∫ (e^(3x)+1)/(e^x+1) dx =
0
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подстановка
e^x=t e^(3x) =t³ e^xdx =dt dx =dt/e^x=dt/t
(x=1 t=e) ( x=0 t=1)
преобразования:
t³+1 (t+1)(t²-t+1)
----- =---------------- = t²-t+1
t+1 (t+1)
(t²-t+1)/t =t-1+1/t
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e e
=∫ (t-1+1/t)dt = (t²/2 -t +lnt) | =(e²/2 -e+1) -(1/2-1+0)=e²/2-e+1,5
1 1
2.8
x=tgt dx=dt/cos²t
√(1+tg²t)³ =√(1+sin²t/cos²t)³ =√((cos²t+sin²t)/cos²t)³ =√(1/cos²t)³=
=1/√(cos²t)³ =1/cos³t
(x=√3 t =π/3) (x=1 t=π/4)
√3 π/3 π/3 π/3
∫ dx/(√(1+x²)³ = ∫ (cos³t/cos²t )dt =∫(cost) dx=∫(cost) dt=
1 π/4 π/4
π/3
= sint | = (√3-√2)/2
π/4