1. ## MTH101 Q1_b

show that the given integral converges or not
∫1/xln^3x dx from e to +∞

2. Originally Posted by TAHIR
show that the given integral converges or not
∫1/xln^3x dx from e to +∞
$\int \frac{1}{x\ln ^3 x} dx$

Let $t=\ln x \implies t' = 1/x$

$\int \frac{1}{t^3} dt = -\frac{1}{t^2}+C = -\frac{1}{\ln^2 x} +C$

Thus,
$\int_e^{\infty} \frac{1}{x\ln^3 x} = \lim_{N\to \infty} - \frac{1}{\ln ^2 N} + e = e$

It converges.

3. Originally Posted by ThePerfectHacker
$\int \frac{1}{x\ln ^3 x} dx$

Let $t=\ln x \implies t' = 1/x$

$\int \frac{1}{t^3} dt = -\frac{1}{t^2}+C = -\frac{1}{\ln^2 x} +C$

Thus,
$\int_e^{\infty} \frac{1}{x\ln^3 x} = \lim_{N\to \infty} - \frac{1}{\ln ^2 N} + e = e$

It converges.