Evaluate the following integrals. Include absolute values only when needed. intfrac{ln^2x+2ln x-1}{x}dx

Integrals
Evaluate the following integrals. Include absolute values only when needed.
$$\int\frac{\ln^2x+2\ln x-1}{x}dx$$

2021-01-14
$$\text{Given integtral, }\int\frac{\ln^2x+2\ln x-1}{x}dx$$
$$\text{we have to find the given integral.}$$
$$\text{here,}\int\frac{\ln^2x+2\ln x-1}{x}dx$$
$$\text{substitute } \ln x=t\Rightarrow\frac{1}{x}\frac{dx}{dt}=1\Rightarrow\frac{1}{x}dx=dt$$
$$\Rightarrow, \int\frac{\ln^2x+2\ln x-1}{x}dx=\int(t^2+2t-1)dt$$
$$=\int t^2dt+2\int tdt-\int 1dt$$
$$=\frac{t^3}{3}+2\frac{t^2}{2}-t+C$$
$$=\frac{t^3}{3}+t^2-t+C$$
$$\text{undo substitution }t=\ln x$$
$$\Rightarrow\int\frac{\ln^2x+2\ln x-1}{x}dx=\frac{\ln^3x}{3}+\ln^2x-\ln x+C$$
$$\text{this is the required answer}$$