Integrals

(∫_0^+*∞)(e^(-x))*d(x)=(∣_0^+*∞)(-e^(-x))=(∣_0^+*∞)(-1/(e^x))=1

(∫_0^+*∞)(e^(-a*x))*d(x)=1/a,a>0

(∫_0^+*∞)(e^(-x^2))*d(x)=√(,π)/2,(∫_0^+*∞)(e^(-a*x^2))*d(x)=1/2√(,π/a)

(∬_R^2^)(e^(-x^2-y^2))*d(x) d(y)=π

(∬_R^2^)(e^(-a*x^2-c*y^2))*d(x) d(y)=π/√(,a*c),a,c<0

(∬_R^2^)(e^(-x^2-2*ρ*x*y-y^2))*d(x) d(y)=π/√(,1-ρ^2),abs(ρ)≤1

(∬_R^2^)(e^(-a*x^2-b*x*y-c*y^2))*d(x) d(y)=π/√(,a*c-b^2),a,c>0

(∫_0^1)((∫_0^1)(1/(1-x*y)))*d(x) d(y)=(π^2)/6,(∬_(0,0)^(1,1))(1/(1-x*y)) d(x) d(y)=(π^2)/6=ζ(2)

(∬_(0,0)^(1,1))(1/(1+x^2*y^2)) d(x) d(y)=(∑_n=0^+*∞)(((-1)^n)/((2*n+1)^2))=𝐺

(∬_(0,0)^(1,1))(1/(1-x^2*y^2)) d(x) d(y)=(∑_n=0^+*∞)(1/((2*n+1)^2))=(π^2)/8

(∬_(0,0)^(+∞,+∞))((e^(-a*x-b*y))/(c*x+d*y))*d(x) d(y)=(ln(a*d)-ln(b*c))/(a*d-b*c)

(∬_(0,0)^(+∞,+∞))((e^(-x-y))/(c*x+d*y))*d(x) d(y)=(ln(d)-ln(c))/(d-c)

(∬_(0,0)^(+∞,+∞))((e^(-a*x-b*y))/(x+y))*d(x) d(y)=(ln(a)-ln(b))/(a-b)

(∬_(-∞,-∞)^(+∞,+∞))((x^(m-1)*y^(n-1)*e^(-x-y))/(x+y))*d(x) d(y)=(Γ(m)*Γ(n))/(m+n-1),m+n>1

(∬_(-∞,-∞)^(+∞,+∞))(1/((x^2+y^2+a^2)^p))*d(x) d(y)=π/((p-1)*a^(2*(p-1))),p>1

(∬_(-∞,-∞)^(+∞,+∞))(1/((x^2+y^2+a^2)^2))*d(x) d(y)=π/(a^2)

(∬_(0,0)^(1,1))|x-y|^p*d(x) d(y)=2/((p+1)*(p+2)),p>-1

(∬_(0,0)^(+∞,+∞))(x^(a-1)*y^(b-1)*e^(-x-y))*d(x) d(y)=Γ(a)*Γ(b)