Question

If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.

Options

• 0.00

• 1.00

• 2.00

• 3.00

Answer 1

If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is 3.00.

Explanation:

Let nx = t

"d"x = `1/"ndt"`

`lim_("n"→∞)(int_("n"/("n" + 1))^1 tan^-1 "tdt")/(int_("n"/("n" + 1))^1 sin^-1 "tdt")`

= `lim_("n"→∞)(-tan^-1("n"/("n" + 1))"d"("n"/("n" + 1)))/(-sin^-1("n"/("n" + 1))"d"("n"/("n" + 1))` 

= `(tan^-1(1))/(sin^-1(1))`

= `(π/4)/(π/2)`

= `1/2`

Now, `"p"/"q" = 1/2`

⇒ p + q = 3