求不定积分:(x*arctanx)/[(1+x^2)^3]
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求不定积分:(x*arctanx)/[(1+x^2)^3]
希望有过程.
希望有过程.
![求不定积分:(x*arctanx)/[(1+x^2)^3]](/uploads/image/z/4121128-64-8.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%EF%BC%9A%28x%2Aarctanx%29%2F%5B%281%2Bx%5E2%29%5E3%5D)
∫(x*arctanx)/[(1+x^2)^3]dx
=∫(1/2)(arctanx)/[(1+x^2)^3]d(x^2+1)
=∫(1/2)(arctanx)(-1/2)d[(x^2+1)^(-2)]
=(-1/4)arctanx/(x^2+1)^2+(1/4)∫(x^2+1)^(-2)d(arctanx)
=(-1/4)arctanx/(x^2+1)^2+(1/4)∫(x^2+1)^(-2)(x^2+1)^2dx
=(-1/4)arctanx/(x^2+1)^2+(1/4)∫dx
=-arctanx/[4(x^2+1)^2]+x/4+C
C是积分任意常数.
=∫(1/2)(arctanx)/[(1+x^2)^3]d(x^2+1)
=∫(1/2)(arctanx)(-1/2)d[(x^2+1)^(-2)]
=(-1/4)arctanx/(x^2+1)^2+(1/4)∫(x^2+1)^(-2)d(arctanx)
=(-1/4)arctanx/(x^2+1)^2+(1/4)∫(x^2+1)^(-2)(x^2+1)^2dx
=(-1/4)arctanx/(x^2+1)^2+(1/4)∫dx
=-arctanx/[4(x^2+1)^2]+x/4+C
C是积分任意常数.
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