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求不定积分∫根号(9+x^2)dx
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分部积分法:
∫√(9+x^2)dx
=x√(9+x^2)dx-∫x^2/√(9+x^2)dx
=x√(9+x^2)dx-∫(9+x^2-9)/√(9+x^2)dx
=x√(9+x^2)dx-∫√(9+x^2)dx+9∫1/√(9+x^2)dx
=x√(9+x^2)dx-∫√(9+x^2)dx+9ln(x+√(9+x^2))
∴∫√(9+x^2)dx=1/2×x√(9+x^2)dx+9/2×ln(x+√(9+x^2))+C
∫√(9+x^2)dx
=x√(9+x^2)dx-∫x^2/√(9+x^2)dx
=x√(9+x^2)dx-∫(9+x^2-9)/√(9+x^2)dx
=x√(9+x^2)dx-∫√(9+x^2)dx+9∫1/√(9+x^2)dx
=x√(9+x^2)dx-∫√(9+x^2)dx+9ln(x+√(9+x^2))
∴∫√(9+x^2)dx=1/2×x√(9+x^2)dx+9/2×ln(x+√(9+x^2))+C
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