Integrate \(\sqrt{4-x^2}\).
\(\dfrac{1}{2}x\sqrt{4-x^2}+2\sin^{-1}\!\left(\dfrac{x}{2}\right)+C\)
Integrate \(\sqrt{1-4x^2}\).
\(\dfrac{1}{4}\sin^{-1}(2x)+\dfrac{1}{2}x\sqrt{1-4x^2}+C\)
Integrate \(\sqrt{x^2+4x+6}\).
\(\dfrac{x+2}{2}\sqrt{x^2+4x+6}+\log\left|x+2+\sqrt{x^2+4x+6}\right|+C\)
Integrate \(\sqrt{x^2+4x+1}\).
\(\dfrac{x+2}{2}\sqrt{x^2+4x+1}-\dfrac{3}{2}\log\left|x+2+\sqrt{x^2+4x+1}\right|+C\)
Integrate \(\sqrt{1-4x-x^2}\).
\(\dfrac{5}{2}\sin^{-1}\!\left(\dfrac{x+2}{\sqrt{5}}\right)+\dfrac{x+2}{2}\sqrt{1-4x-x^2}+C\)
Integrate \(\sqrt{x^2+4x-5}\).
\(\dfrac{x+2}{2}\sqrt{x^2+4x-5}-\dfrac{9}{2}\log\left|x+2+\sqrt{x^2+4x-5}\right|+C\)
Integrate \(\sqrt{1+3x-x^2}\).
\(\dfrac{2x-3}{4}\sqrt{1+3x-x^2}+\dfrac{13}{8}\sin^{-1}\!\left(\dfrac{2x-3}{\sqrt{13}}\right)+C\)
Integrate \(\sqrt{x^2+3x}\).
\(\dfrac{2x+3}{4}\sqrt{x^2+3x}-\dfrac{9}{8}\log\left|x+\dfrac{3}{2}+\sqrt{x^2+3x}\right|+C\)
Integrate \(\sqrt{1+\dfrac{x^2}{9}}\).
\(\dfrac{x}{6}\sqrt{x^2+9}+\dfrac{3}{2}\log\left|x+\sqrt{x^2+9}\right|+C\)
\(\int \sqrt{1+x^2}\,dx\) is equal to
\(\dfrac{x}{2}\sqrt{1+x^2}+\dfrac{1}{2}\log\left|x+\sqrt{1+x^2}\right|+C\)
\(\dfrac{2}{3}(1+x^2)^{3/2}+C\)
\(\dfrac{2}{3}x(1+x^2)^{3/2}+C\)
\(\dfrac{x^2}{2}\sqrt{1+x^2}+\dfrac{1}{2}x^2\log\left|x+\sqrt{1+x^2}\right|+C\)
\(\int \sqrt{x^2-8x+7}\,dx\) is equal to
\(\dfrac{1}{2}(x-4)\sqrt{x^2-8x+7}+9\log\left|x-4+\sqrt{x^2-8x+7}\right|+C\)
\(\dfrac{1}{2}(x+4)\sqrt{x^2-8x+7}+9\log\left|x+4+\sqrt{x^2-8x+7}\right|+C\)
\(\dfrac{1}{2}(x-4)\sqrt{x^2-8x+7}-3\sqrt{2}\,\log\left|x-4+\sqrt{x^2-8x+7}\right|+C\)
\(\dfrac{1}{2}(x-4)\sqrt{x^2-8x+7}-\dfrac{9}{2}\log\left|x-4+\sqrt{x^2-8x+7}\right|+C\)