Question

Evaluate the following:

`int_0^1 (sin(3tan^-1 x)tan^-1 x)/(1 + x^2)  "d"x`

Answer 1

I = `int_0^1 (sin(3tan^-1 x)tan^-1 x)/(1 + x^2)  "d"x`

Put t = `tan^-1x`

dt = `1/(1 + x^2)  "d"x`

x	0	1
t	0	`pi/4`

I = `int_0^(pi/4) (sin 3"t")"t"  "dt"`

= `int_0^(pi/4) "t"(sin 3"t")  "dt"`

u = t, v = sin3t

u' = 1, v₁ = `- (cos3"t")/3`

u'' = 0, v₁ = `- (sin 3"t")/9` 

`int "uv"  "d"x` = uv₁ – u'v₂ + u"v₃ 

`int_0^(pi/) "t"(sin 3"t")  "dt" = [- "t" (cos3"t")/3 + (sin 3"t")/9]_0^(pi/4)`

= `-^((pi/4)) xx 1/3 xx - 1/sqrt(2) + 1/9 xx 1/sqrt(2)`

= `pi/(12sqrt(2)) + 1/(9sqrt(2)`

= `1/sqrt(2) [pi/12 + 1/9]`