Methods of Integration

(cod: P-70-71-15) Consider the following questions:

(1) Calculate \(\displaystyle \int \dfrac{1}{x \ln^2{x}}\, dx\).

(2) What can be stated about the improper integral \(\displaystyle \int_0^{+ \infty}x e^{-x^2}\, dx\)?

(3) Select an integral that has the same value as \(\displaystyle \int_0^{\frac{\pi}{2}} e^{\sin\, {x}}\, \sin(2x)\, dx\).

(4) Determine \(\displaystyle \int \tan{x}\, dx\).

(1) \(-\dfrac{1}{\ln{x}} + C.\)

(2) \(\displaystyle \int_0^{+\infty} xe^{-x^2}\, dx = 1/2\)

(3) \(\displaystyle 2\int_0^1 ue^u \, du\)

(4) \(\ln|\sec{x}| + C\)

(1) \(\dfrac{1}{\ln{x}} + C.\)

(2) \(\displaystyle \int_0^{+\infty} xe^{-x^2}\, dx = 1\)

(3) \(\displaystyle \int_0^1 ue^u \, du\)

(4) \(-\ln|\sec{x}| + C\)

(1) \(-\dfrac{1}{\ln{x}} + C.\)

(2) \(\displaystyle \int_0^{+\infty} xe^{-x^2}\, dx = 1/2\)

(3) \(\displaystyle \int_0^1 ue^u \, du\)

(4) \(\ln|\cos{x}| + C\)

(1) \(-\dfrac{1}{\ln{x}} + C.\)

(2) \(\displaystyle \int_0^{+\infty} xe^{-x^2}\, dx = 1\)

(3) \(\displaystyle 2\int_0^1 e^u \, du\)

(4) \(-\ln|\cos{x}| + C\)

(1) \(\dfrac{1}{\ln{x}} + C.\)

(2) \(\displaystyle \int_0^{+\infty} xe^{-x^2}\, dx\) is divergent.

(3) \(\displaystyle 2\int_0^1 ue^u \, du\)

(4) \(\ln|\sec{x}| + C\)