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Integral with ±ax2+bx+c\sqrt{\pm ax^2+bx+c}

73​

∫1ax2+bx+c dx=1aln⁡∣2ax+b+2aax2+bx+c∣+C\int{\frac{1}{\sqrt{ax^2+bx+c}}}\ \mathrm{d}x=\frac{1}{\sqrt{a}}\ln{\left|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c} \right|}+C
Derivation

74​

∫ax2+bx+c dx=2ax+b4aax2+bx+c+4ac−b28a3ln⁡∣2ax+b+2aax2+bx+c∣+C\int{\sqrt{ax^2+bx+c}}\ \mathrm{d}x=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c}+\frac{4ac-b^2}{8\sqrt{a^3}}\ln{\left|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c} \right|}+C
Derivation

75​

∫xax2+bx+c dx=1aax2+bx+c−b2a3ln⁡∣2ax+b+2aax2+bx+c∣+C\int{\frac{x}{\sqrt{ax^2+bx+c}}}\ \mathrm{d}x=\frac{1}{a}\sqrt{ax^2+bx+c}-\frac{b}{2\sqrt{a^3}}\ln{\left|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c} \right|}+C
Derivation

76​

∫1−ax2+bx+c dx=1aarcsin⁡2ax−bb2+4ac+C\int{\frac{1}{\sqrt{-ax^2+bx+c}}}\ \mathrm{d}x=\frac{1}{\sqrt{a}}\arcsin{\frac{2ax-b}{\sqrt{b^2+4ac}}}+C
Derivation

77​

∫−ax2+bx+c dx=2ax−b4a−ax2+bx+c+b2+4ac8a3arcsin⁡2ax−bb2+4ac+C\int{\sqrt{-ax^2+bx+c}}\ \mathrm{d}x=\frac{2ax-b}{4a}\sqrt{-ax^2+bx+c}+\frac{b^2+4ac}{8\sqrt{a^3}}\arcsin{\frac{2ax-b}{\sqrt{b^2+4ac}}} +C
Derivation

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∫x−ax2+bx+c dx=−1a−ax2+bx+c+b2a3arcsin⁡2ax−bb2+4ac+C\int{\frac{x}{\sqrt{-ax^2+bx+c}}}\ \mathrm{d}x=-\frac{1}{a}\sqrt{-ax^2+bx+c}+\frac{b}{2\sqrt{a^3}}\arcsin{\frac{2ax-b}{\sqrt{b^2+4ac}}}+C
Derivation