Question

Evaluate the integral $\displaystyle{\int_{1}^{e}(\ln{x})^{2}dx}∫$ using the integration by parts twice

1. e-2
2. e+2
3. e
4. 1/e

Answer 1

$\int_1^e (lnx)^2 dx\\ put\\ lnx=t\implies x=e^t\\ dx=e^tdt\\ Therefore,\\ \int_1^e (lnx)^2 dx=\int_0^1 t^2 e^t dt\\ \text{By using by parts method, we get}\\ =t^2\int e^tdt-\int [\frac{dt^2}{dt} \int e^tdt]dt\\ =t^2e^t-2\int te^tdt\\ \text{By using by parts method, we get}\\ =t^2e^t-2[te^t-e^t]\\ =[t^2e^t-2te^t+2e^t]_0^1\\ =e-2e+2e-2\\ =e-2$

Therefore, option 1 is correct.