∫ LN(x^2 + 3) dx
∫ 1·LN(x^2 + 3) dx
Partielle Integration
x·LN(x^2 + 3) - ∫ x·(2·x)/(x^2 + 3) dx
x·LN(x^2 + 3) - ∫ 2·x^2/(x^2 + 3) dx
x·LN(x^2 + 3) - ∫ (2 - 6/(x^2 + 3)) dx
x·LN(x^2 + 3) - ∫ 2 dx + ∫ 6/(x^2 + 3) dx
x·LN(x^2 + 3) - 2·x + 2·√3·arctan(x/√3) + c
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∫ 6/(x^2 + 3) dx
∫ 2/(x^2/3 + 1) dx
Subst. z = x/√3
dz = 1/√3 dx --> dx = √3 dz
∫ 2/(z^2 + 1) √3 dz
∫ 2·√3 · 1/(z^2 + 1) dz
2·√3·arctan(z) + c
Resubst.
2·√3·arctan(x/√3) + c