求不定积分∫x^2/根号下(x^2+a^2) dx (a>0)
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求不定积分∫x^2/根号下(x^2+a^2) dx (a>0)
![求不定积分∫x^2/根号下(x^2+a^2) dx (a>0)](/uploads/image/z/3909884-68-4.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E2%88%ABx%5E2%2F%E6%A0%B9%E5%8F%B7%E4%B8%8B%28x%5E2%2Ba%5E2%29+dx+%28a%3E0%29)
∫x^2/√(a^2+x^2)dx
=∫(x^2+a^2-a^2)/√(a^2+x^2)dx
=∫√(x^2+a^2)dx-a^2∫dx/√(a^2+x^2)
=x√(x^2+a^2)- ∫x√d(x^2+a^2)dx-a^2arsh(x/a)
= x√(x^2+a^2)- ∫x^2dx/√(x^2+a^2)-a^2(ln(x/a+√(1+(x/a)^2)),
2∫x^2dx/√(x^2+a^2)= x√(x^2+a^2)-a^2{ln[x+√(a^2+x^2)]},
∴∫x^2dx/√(a^2+x^2)= x√(a^2+x^2)/2-a^2ln[x+√(a^2+x^2)]/2+C
这里用到分部积分和反双曲正弦函数arshx.
=∫(x^2+a^2-a^2)/√(a^2+x^2)dx
=∫√(x^2+a^2)dx-a^2∫dx/√(a^2+x^2)
=x√(x^2+a^2)- ∫x√d(x^2+a^2)dx-a^2arsh(x/a)
= x√(x^2+a^2)- ∫x^2dx/√(x^2+a^2)-a^2(ln(x/a+√(1+(x/a)^2)),
2∫x^2dx/√(x^2+a^2)= x√(x^2+a^2)-a^2{ln[x+√(a^2+x^2)]},
∴∫x^2dx/√(a^2+x^2)= x√(a^2+x^2)/2-a^2ln[x+√(a^2+x^2)]/2+C
这里用到分部积分和反双曲正弦函数arshx.