作业帮定积分:分母是X乘上根号下(x^2+4),分子是一.上限是2根号3,下限是2
来源：学生作业帮助网 编辑：作业帮 时间：2024/11/15 08:19:12
定积分:分母是X乘上根号下(x^2+4),分子是一.上限是2根号3,下限是2
定积分:分母是X乘上根号下(x^2+4),分子是一.上限是2根号3,下限是2
定积分:分母是X乘上根号下(x^2+4),分子是一.上限是2根号3,下限是2
换元法:
x=2tanθ,由于x∈[2,2√3],从而θ∈[π/4,π/3],
√(x²+4)=2/cosθ,dx=2dθ/cos²θ,
从而∫[2,2√3]dx/(x√(x²+4))
=1/2∫[π/4,π/3]dθ/sinθ
=-1/4*ln2+1/2*ln(2+√2)-1/4*ln3
凑微分法:
∫[2,2√3]dx/(x√(x²+4))
=∫[2,2√3]dx/(x²√(1+4/x²))
=∫[2,2√3]x^(-2)dx/√(1+4/x²)
=-1/2∫[2,2√3]d(2/x)/√(1+(2/x)²)
=-1/2ln(2/x+√(1+(2/x)²))[2,2√3]
=-1/4*ln2+1/2*ln(2+√2)-1/4*ln3
换元 x= 2 tga
利用 (tg a)^2 + 1=(sec a)^2
tg a= sin a/cos a
两个式子带进去凑微分。
根据公式∫1/(a^2+x^2)dx=(1/a)*arctan(x/a)+c
∫1/(2^2+x^2)dx=(1/2)*arctan(x/2)+c
其定积分=(1/2)*arctan(x/2)|2√3-(1/2)*arctan(x/2)|2
=(1/2)*arctan√3-(1/2)*arctan1
=(1/2)(mπ+π/6)-(1/2)(kπ+π/4)
=(1/2)(nπ-π/12)
∵设x=2tant,则当x=2√3时，t=π/3.当x=2时，t=π/4
∴原式=1/2∫(π/4,π/3)dt/sint (∫(π/4,π/3)表示从π/4到π/3积分)
=1/2∫(π/4,π/3)sintdt/sin²t
=1/2∫(π/4,π/3)d(cost)/(cos²t-1)
...
全部展开
∵设x=2tant,则当x=2√3时，t=π/3.当x=2时，t=π/4
∴原式=1/2∫(π/4,π/3)dt/sint (∫(π/4,π/3)表示从π/4到π/3积分)
=1/2∫(π/4,π/3)sintdt/sin²t
=1/2∫(π/4,π/3)d(cost)/(cos²t-1)
=1/4∫(π/4,π/3)[1/(cost-1)-1/(cost+1)]d(cost)
=1/4[ln|cost-1|-ln|cost+1|]|(π/4,π/3)
=1/4[ln|(cost-1)/(cost+1)|]|(π/4,π/3)
=1/4[ln(1/3)+2ln(√2+1)] （代值化简）
=1/4ln(1+2√2/3).
收起