# Calculus 2

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Submitted By Jdyoung
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\int f'(x)e^{{f(x)}}\;{\mathrm {d}}x=e^{{f(x)}}
\int e^{{cx}}\;{\mathrm {d}}x={\frac {1}{c}}e^{{cx}}
\int a^{{cx}}\;{\mathrm {d}}x={\frac {1}{c\cdot \ln a}}a^{{cx}} for a>0,\ a\neq 1
\int xe^{{cx}}\;{\mathrm {d}}x={\frac {e^{{cx}}}{c^{2}}}(cx-1)
\int x^{2}e^{{cx}}\;{\mathrm {d}}x=e^{{cx}}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)
\int x^{n}e^{{cx}}\;{\mathrm {d}}x={\frac {1}{c}}x^{n}e^{{cx}}-{\frac {n}{c}}\int x^{{n-1}}e^{{cx}}{\mathrm {d}}x=\left({\frac {\partial }{\partial c}}\right)^{n}{\frac {e^{{cx}}}{c}}
\int {\frac {e^{{cx}}}{x}}\;{\mathrm {d}}x=\ln |x|+\sum _{{n=1}}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}
\int {\frac {e^{{cx}}}{x^{n}}}\;{\mathrm {d}}x={\frac {1}{n-1}}\left(-{\frac {e^{{cx}}}{x^{{n-1}}}}+c\int {\frac {e^{{cx}}}{x^{{n-1}}}}\,{\mathrm {d}}x\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}
\int e^{{cx}}\ln x\;{\mathrm {d}}x={\frac {1}{c}}\left(e^{{cx}}\ln |x|-\operatorname {Ei}\,(cx)\right)
\int e^{{cx}}\sin bx\;{\mathrm {d}}x={\frac {e^{{cx}}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)
\int e^{{cx}}\cos bx\;{\mathrm {d}}x={\frac {e^{{cx}}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)
\int e^{{cx}}\sin ^{n}x\;{\mathrm {d}}x={\frac {e^{{cx}}\sin ^{{n-1}}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{{cx}}\sin ^{{n-2}}x\;{\mathrm {d}}x
\int e^{{cx}}\cos ^{n}x\;{\mathrm {d}}x={\frac {e^{{cx}}\cos ^{{n-1}}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{{cx}}\cos ^{{n-2}}x\;{\mathrm {d}}x
\int xe^{{cx^{2}}}\;{\mathrm {d}}x={\frac {1}{2c}}\;e^{{cx^{2}}}
\int e^{{-cx^{2}}}\;{\mathrm {d}}x={\sqrt {{\frac {\pi }{4c}}}}\operatorname {erf}({\sqrt {c}}x) (\operatorname {erf} is the error function)
\int xe^{{-cx^{2}}}\;{\mathrm {d}}x=-{\frac {1}{2c}}e^{{-cx^{2}}}
\int {\frac {e^{{-x^{2}}}}{x^{2}}}\;{\mathrm {d}}x=-{\frac

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