Question

Evaluate the following integrals using properties of integration:

`int_0^1 (log(1 + x))/(1 + x^2)  "d"x`

Answer 1

Let I = `int_0^1 (log(1 + x))/(1 + x^2)  "d"x`

Now, on putting x = tanθ

dx = sec²θ dθ

x	0	1
θ	0	`pi/4`

I = `int_0^(pi/4) (log(1 + tan theta))/(1 + tan^2theta) sec^2 theta  "d"theta`

I = `int_0^(pi/4) (log(1 + tan theta))/(sec^2theta) sec^2theta  "d"theta`

`int_0^"a" f(x)  "d"x = int_0^"a" f("a" - x)  "d"x`

I = `int_0^(pi/4) log(1 + tan(pi/4 - theta)) "d"theta`

= `int_0^(pi/4) log[1 + (tan  pi/4 - tan theta)/(1 + tan  pi/4 tan theta)] "d"theta`

= `int_0^(pi/4)  log(1 + (1 - tan theta)/(1 + tan theta))"d"theta`

= `int_0^(pi/4) log ((1 + tan theta + 1 - tan theta)/(1 + tan theta)) "d"theta`

= `int_0^(pi/4) log (2/(1 + tan theta)) "d"theta`

= `int_0^(pi/4) [log2 - log(1 + tan theta)] "d"theta`

= `log 2 int_0^(pi/4) "d"theta - int_0^(pi/4) log(1 + tan theta)  "d"theta`

I = `log 2(pi/4) - "I"`

2I = `pi/4 log 2`

I = `pi/8 log 2`