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

Question
Integrals
asked 2021-01-13
Evaluate the following integrals. Include absolute values only when needed.
\(\int\frac{\ln^2x+2\ln x-1}{x}dx\)

Answers (1)

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