Монгол Бодлогын Сан

Заавар:$$\int f(x)\,\mathrm{d}x=F(x)+C$$ бол $$\int_a^b f[g(x)]g^\prime(x)\,\mathrm{d}x=\int_a^b f[g(x)]\,\mathrm{d}g(x)=$$ $$=\int_{g(a)}^{g(b)} f(t)\,\mathrm{d}t=F[g(b)]-F[g(a)]$$ байдаг. Үүнийг тодорхой биш интегралыг бодох орлуулгын арга гэдэг.

Бодолт:$$\displaystyle\int_{0}^{\ln 3}\dfrac{e^{x}}{e^{x} +1} \,\mathrm{d}x=\int_0^{\ln 3}\dfrac{d(e^x+1)}{e^x+1}\overset{[t=e^x+1]}{=\kern-0.1em=\kern-0.1em=\kern-0.1em=}\int_{e^0+1}^{e^{\ln 3}+1}\dfrac{dt}{t}$$ $$\int_{e^0+1}^{e^{\ln 3}+1}\dfrac{dt}{t}=\int_{2}^{4}\dfrac{dt}{t}=\ln t\bigg|_{2}^{4}=\ln 4-\ln 2=\ln\dfrac{4}{2}=\ln2$$