What is the integral of ((ln(t))/t)^2 ?

The integral of ((ln(t))/t)^2 is -(ln^2(t))/t+2(-(ln(t))/t-1/t)+C

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integral of ((ln(t))/t)^2

integral

(\frac{ln ( t )}{t})^{2}

- \frac{ln ^{2} ( t )}{t} + 2 (- \frac{ln ( t )}{t} - \frac{1}{t}) + C

Solution steps

\int (\frac{ln ( t )}{t})^{2} d t

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \int \frac{ln ^{2} ( t )}{t ^{2}} d t

Apply Integration By Parts

: - \frac{ln ^{2} ( t )}{t} - \int - \frac{2 ln ( t )}{t ^{2}} d t

= - \frac{ln ^{2} ( t )}{t} - \int - \frac{2 ln ( t )}{t ^{2}} d t

\int - \frac{2 ln ( t )}{t ^{2}} d t = - 2 (- \frac{ln ( t )}{t} - \frac{1}{t})

= - \frac{ln ^{2} ( t )}{t} - (- 2 (- \frac{ln ( t )}{t} - \frac{1}{t}))

Simplify

= - \frac{ln ^{2} ( t )}{t} + 2 (- \frac{ln ( t )}{t} - \frac{1}{t})

Add a constant to the solution

= - \frac{ln ^{2} ( t )}{t} + 2 (- \frac{ln ( t )}{t} - \frac{1}{t}) + C

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What is the integral of ((ln(t))/t)^2 ?
The integral of ((ln(t))/t)^2 is -(ln^2(t))/t+2(-(ln(t))/t-1/t)+C