Integralrechnungen
partielle Integration
Integral(u * dv) = u * v - Integral(v * du)Integral(u * v' * dx) = u * v - Integral(u' * v * dx)
Integral(x^n*e^x)=x^n*e^x-n*Integral((x^(n-1))*e^x)
Integral(x^n*sin(x)*dx)=-x^n*cos(x)+n*Integral((x^(n-1))*cos(x)*dx)
Integral(x^n*cos(x)*dx)=+x^n*sin(x)-n*Integral((x^(n-1))+sin(x)*dx)
Integral(ln(x)^n*dx)=x*(ln(x)^n)-n*Integral(ln(x)^(n-1)*dx) n ganz, n != -1
Integral(x^n*ln(x)*dx)=ln(x)*(x^(n+1))/(n+1)-Integral((x^n/(n+1))*dx)=((x^(n+1))/(n+1))*(ln(x)-(1/(n+1))) n != -1
Integral(sin^nx*dx)=-((cos(x)*sin^(n-1)*x)/n)+((n-1)/n)*Integral(sin^(n-2)x*dx) n ganz
Integral(cos^nx*dx)=((sin(x)*cos^(n-1)*x)/n)+((n-1)/n)*Integral(cos^(n-2)x*dx) n !=0
Substitutionsmethode
Integral((fi'(x)/fi(x))*dx)=ln|fi(x)|+CIntegral((x^(n-1)/(x^n+a))*dx)=(1/n)*ln|x^n+a|+C
Integral(dx/(sqrt(x^2+a^1)))=ln|x+sqrt(x^2+a^2)|+C
Integral(tan(x)*dx)=-ln|cos(x)|+C
Integral(cot(x)*dx)=ln|sin(x)|+C
Integral(tanh(x)*dx)=ln|cosh(x)|+C
Integral(coth(x)*dx)=ln|sinh(x)|+C
Integral(arctan(x)*dx)=x*arctan(x)-ln|sqrt(1+x^2)|+C
Integral(arccot(x)*dx)=x*arccot(x)+ln|sqrt(1+x^2)|+C
Integral(artanh(x)*dx)=x*artanh(x)+ln|sqrt(1-x^2)|+C, |x| < 1
Integral(arcoth(x)*dx)=x*arcoth(x)+ln|sqrt(x^2-1)|+C, |x| > 1
Integral(fi(x)fi'(x)*dx=(1/2)*[fi(x)]^2+C
Integral(fi^n(x)*fi'(x)*dx=Integral(z^n*dz=(1/(n+1))*fi^(n+1)(x)+C
Integral(arcsin(x)*dx)=x*arcsin(x)+sqrt(1-x^2)+C
Integral(arccos(x)*dx)=x*arccos(x)-sqrt(1-x^2)+C
Integral(arsinh(x)*dx)=x*arsinh(x)-sqrt(1+x^2)+C
Integral(arcosh(x)*dx)=x*arcosh(x)-sqrt(x^2-1)+C
Integral(dx/sqrt(a^2-x^2))=arcsin(x/a)+C
Integral(dx/(a^2+x^2))=(1/a)*arctan(x/a)+C
Integral(dx/sqrt(a^2-b^2*x^2))=(1/b)*arcsin(b/a)*x+C
Integral(dx/(a^2+b^2*x^2)=(1/(a*b))*arctan(b/a)*x+C
