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arctane^x/e^2xdx
题目详情
arctane^x/e^2xdx
▼优质解答
答案和解析
令e^x=t x=lnt
∫arctane^x/e^2xdx=∫arctant/t^3dt=-1/2∫arctantd(1/t^2)
=-1/2[(arctant/t^2)-∫1/(t^2)(1+t^2)dt]
=-1/2{(arctant/t^2)-∫[1/(t^2)]-[1/(1+t^2)]dt}
=-1/2[(arctant/t^2)+1/t+arctant+c]
=-1/2[(arctane^x/e^2x)+1/e^x+arctane^x+c]
∫arctane^x/e^2xdx=∫arctant/t^3dt=-1/2∫arctantd(1/t^2)
=-1/2[(arctant/t^2)-∫1/(t^2)(1+t^2)dt]
=-1/2{(arctant/t^2)-∫[1/(t^2)]-[1/(1+t^2)]dt}
=-1/2[(arctant/t^2)+1/t+arctant+c]
=-1/2[(arctane^x/e^2x)+1/e^x+arctane^x+c]
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