(九) 含有 $ \sqrt{\pm ax^2+bx+c} \quad (a>0) $的积分

$$ 73.\,\int\!\! \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}\ln \vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\,\vert +C $$

$$ 74.\,\int\!\! \sqrt{ax^2+bx+c}\,dx=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c} $$

$$ \qquad +\frac{4ac-b^2}{8\sqrt{a^3}}\ln\vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\vert +C $$

$$ 75.\,\int\!\! \frac{x}{\sqrt{ax^2 + bx + c}}dx = \frac{1}{a}\sqrt{ ax^2 + bx + c} $$

$$ \qquad -\frac{b}{2\sqrt{a^3}}\ln \vert 2ax + b + 2 \sqrt{a}\sqrt{ax^2 + bx + c}\,\vert +C $$

$$ 76.\,\int\!\! \frac{dx}{\sqrt{c+bx-ax^2}}= -\frac{1}{\sqrt{a}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C $$

$$ 77.\,\int\!\! \sqrt{c+bx-ax^2}\,dx=\frac{2ax-b}{4a}\sqrt{c+bx-ax^2}+\frac{b^2+4ac}{8\sqrt{a^3}}\arcsin \frac{2ax-b}{\sqrt{b^2+4ac}}+C $$

$$ 78.\,\int\!\! \frac{x}{\sqrt{c+bx-ax^2}}\,dx=-\frac{1}{a}\sqrt{c+bx-ax^2}+\frac{b}{2\sqrt{a^3}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C $$

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