Daftar integral dari fungsi eksponensial


Dari Wikipedia Indonesia, artikel bebas.

 

int e^{cx};dx = frac{1}{c} e^{cx}
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int xe^{cx}; dx = frac{e^{cx}}{c^2}(cx-1)
int x^2 e^{cx};dx = e^{cx}left(frac{x^2}{c}-frac{2x}{c^2}+frac{2}{c^3}right)
int x^n e^{cx}; dx = frac{1}{c} x^n e^{cx} - frac{n}{c}int x^{n-1} e^{cx} dx
intfrac{e^{cx}}{x}; dx = ln|x| +sum_{i=1}^inftyfrac{(cx)^i}{icdot i!}
intfrac{e^{cx}}{x^n}; dx = frac{1}{n-1}left(-frac{e^{cx}}{x^{n-1}}+cintfrac{e^{cx} }{x^{n-1}},dxright) qquadmbox{(untuk }nneq 1mbox{)}
int e^{cx}ln x; dx = frac{1}{c}e^{cx}ln|x|-operatorname{Ei},(cx)
int e^{cx}sin bx; dx = frac{e^{cx}}{c^2+b^2}(csin bx - bcos bx)
int e^{cx}cos bx; dx = frac{e^{cx}}{c^2+b^2}(ccos bx + bsin bx)
int e^{cx}sin^n x; dx = frac{e^{cx}sin^{n-1} x}{c^2+n^2}(csin x-ncos x)+frac{n(n-1)}{c^2+n^2}int e^{cx}sin^{n-2} x;dx
int e^{cx}cos^n x; dx = frac{e^{cx}cos^{n-1} x}{c^2+n^2}(ccos x+nsin x)+frac{n(n-1)}{c^2+n^2}int e^{cx}cos^{n-2} x;dx
int x e^{c x^2 }; dx= frac{1}{2c} ;  e^{c x^2}
int e^{-c x^2 }; dx= sqrt{frac{pi}{4c}} mbox{erf}(sqrt{c} x)(erf adalah fungsi kesalahan/error function)
int xe^{-c x^2 }; dx=-frac{1}{2c}e^{-cx^2}
int {1 over sigmasqrt{2pi} },e^{-{(x-mu )^2 / 2sigma^2}}; dx= frac{1}{2} (1 + mbox{erf},frac{x-mu}{sigma sqrt{2}})
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dimana  c_{2j}=frac{ 1 cdot 3 cdot 5 cdots (2j-1)}{2^{j+1}}=frac{(2j),!}{j!, 2^{2j+1}}  .

 

Integral Tertentu

 

0)” /> (integral Gauss)
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-1,a>0)
frac{(2k-1)!!}{2^{k+1}a^k}sqrt{frac{pi}{a}} & (n=2k, k ;text{bilangan bulat}, a>0)
frac{k!}{2a^{k+1}} & (n=2k+1,k ;text{bilangan bulat}, a>0)
end{cases} ” />
-1,a>0)
frac{n!}{a^{n+1}} & (n=0,1,2,ldots,a>0)
end{cases}” />
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int_{0}^{2 pi} e^{x cos theta} d theta = 2 pi I_{0}(x) (I0 adalah perubahan dari fungsi Bessel jenis pertama)
int_{0}^{2 pi} e^{x cos theta + y sin theta} d theta = 2 pi I_{0} left( sqrt{x^2 + y^2} right)
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