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प्रश्न
Evaluate the following integrals as the limits of sum:
`int_0^1 (5x + 4)"d"x`
उत्तर
Here f(x) = 5x + 4
a = 0
b = 1
Hence we get
`int_"a"^"b" f(x)"d"x = lim_("n" -> oo) 1/"n" sum_("r" = 1)^"n" f("r"/"n")`
⇒ `int_0^1 (5x + 4)"d"x = lim_("n" -> oo) 1/"n" [sum_("r" -> 1)^"n" (5("r"/"n") + 4)]`
= `lim_("n" -> oo) 1/"n" (sum_("r" = 1)^"n" 5("r"/"n") + sum_("r" = 1)^"n" 4)`
= `lim_("n" -> oo) 1/"n" (5/"n" sum_("r" = 1)^"n" "r" + sum_("r" = 1)^"n" 4)`
= `lim_("n" ->oo) 1/"n" (5/"n" xx ("n"("n" + 1))/2 + 4"n")`
= `lim_("n" -> oo) 1/"n" ((5"n" + 5 + 8"n")/2)`
= `lim_("n" -> oo) 1/"n" ((13"n" + 5)/2)`
= `lim_("n" -> oo) 1/"n" xx "n"/2 (13 + 5/"n")`
= `13/2`
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