求不定积分∫[x·arctanx/﹙1+x^2)]dx,
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求不定积分∫[x·arctanx/﹙1+x^2)]dx,
打错了,是求不定积分∫[x·arctanx/﹙1+x^2)^2]dx,
打错了,是求不定积分∫[x·arctanx/﹙1+x^2)^2]dx,
![求不定积分∫[x·arctanx/﹙1+x^2)]dx,](/uploads/image/z/5426600-32-0.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E2%88%AB%5Bx%C2%B7arctanx%EF%BC%8F%EF%B9%991%EF%BC%8Bx%5E2%29%5Ddx%2C)
∫ [ x·arctanx/(1+x^2)^2] dx 换元 u=arctanx,x=tanu,dx=(secu)^2 du
= ∫ tanu * u / (secu)^2 du = (1/2) ∫ u sin(2u) du
= (-1/4) ∫ u d cos(2u) 分部积分
= (-1/4) [ u cos(2u) - (1/2)sin(2u) ] + C
= (-1/4) arctanx * (1-x²)/(1+x²) + (1/4) x /(1+x²) + C
= ∫ tanu * u / (secu)^2 du = (1/2) ∫ u sin(2u) du
= (-1/4) ∫ u d cos(2u) 分部积分
= (-1/4) [ u cos(2u) - (1/2)sin(2u) ] + C
= (-1/4) arctanx * (1-x²)/(1+x²) + (1/4) x /(1+x²) + C
求不定积分∫[x·arctanx/﹙1+x^2)]dx,
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