Check the solution to "Evaluate the following integrals. Include absolute values only..."

melodykap

Answered question

2021-01-13

Evaluate the following integrals. Include absolute values only when needed.

\int \frac{\ln^{2} x + 2 \ln x - 1}{x} d x

SoosteethicU

Skilled2021-01-14Added 102 answers

Given integtral, \int \frac{\ln^{2} x + 2 \ln x - 1}{x} d x

we have to find the given integral.

here, \int \frac{\ln^{2} x + 2 \ln x - 1}{x} d x

substitute \ln x = t \Rightarrow \frac{1}{x} \frac{d x}{d t} = 1 \Rightarrow \frac{1}{x} d x = d t

\Rightarrow, \int \frac{\ln^{2} x + 2 \ln x - 1}{x} d x = \int (t^{2} + 2 t - 1) d t

= \int t^{2} d t + 2 \int t d t - \int 1 d t

= \frac{t^{3}}{3} + 2 \frac{t^{2}}{2} - t + C

= \frac{t^{3}}{3} + t^{2} - t + C

undo substitution t = \ln x

\Rightarrow \int \frac{\ln^{2} x + 2 \ln x - 1}{x} d x = \frac{\ln^{3} x}{3} + \ln^{2} x - \ln x + C

this is the required answer

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