积分唯一性定理与换元法

积分唯一性定理

F(x)=G(x)+c

(F-G)^\prime=F^\prime-G^\prime=0\\{}\\ F(x)-G(x)=c

换元法（变量代换）

\displaystyle \int x^3(x^4+2)^5dx\\{}\\ u=x^4+2\Rightarrow du=4x^3dx\\{}\\ \int \underbrace{(x^4+2)^5}_{u^5}\underbrace{x^3dx}_{\frac{1}{4}du}=\int\frac{u^5du}{4}\\{}\\ =\frac{1}{24}u^6+c\\{}\\ =\frac{1}{24}(x^4+2)^6+c\\{}\\ \int\frac{xdx}{\sqrt{1+x^2}}\\{}\\ u=1+x^2\Rightarrow du=2xdx\\{}\\ ...\int\frac{xdx}{\sqrt{1+x^2}}=\sqrt{1+x^2}+c\\{}\\ \int e^{6x}dx=\frac{1}{6}e^{6x}+c\\{}\\ \int xe^{-x^2}dx=-\frac{1}{2}e^{-x^2}+c

\displaystyle \int\sin x\cos xdx=\frac{1}{2}\sin^2x+c

\displaystyle \frac{d}{dx}\sin^2x=2\sin x\cos x

\displaystyle \frac{d}{dx}\cos^2x=2\cos x(-\sin x)

\displaystyle \int\sin x\cos xdx=-\frac{1}{2}\cos^2x+c

\displaystyle \int\sin x\cos xdx=\frac{1}{2}\sin^2x+c_1\\{}\\ \int\sin x\cos xdx=-\frac{1}{2}\cos^2x+c_2

\displaystyle \frac{1}{2}\sin^2x-(-\frac{1}{2}\cos^2x)\\{}\\ =\frac{1}{2}(\sin^2x+\cos^2x)=\frac{1}{2}\\{}\\ \therefore c_1-c_2=-\frac{1}{2}

\displaystyle \int\frac{dx}{x\ln x}

\displaystyle u=\ln x\\{}\\ du=\frac{dx}{x}\\{}\\ \int\frac{dx}{x\ln x}=\int\underbrace{\frac{1}{\ln x}}_{\frac{1}{u}}\underbrace{\frac{dx}{x}}_{du}\\{}\\ =\int\frac{du}{u}\\{}\\ =\ln u+c\\{}\\ =\ln(\ln x)+c

\displaystyle =\ln|\ln x|+c