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प्रश्न
Evaluate the following integrals as the limit of the sum:
`int_0^1 (x + 4) "d"x`
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उत्तर
We have `int_"a"^"b" "f"(x) "d"x = lim_(("n" -> oo),("h" -> 0)) sum_("r" = 1)^"n" "hf" ("a" + "rh")`
Here a = 0
b = 1
h = `("b" - "a")/"n"`
= `(1 - 0)/"n"`
= `1/"n"`
f(x) = x + 4
f(a + rh) = `"f"(0 + "r"/"n")`
= `"f"("r"/"n")`
`int "f"(x) "d"x = lim_("n" -> oo) sum_("r" = 1)^"n" 1/"n" ("r"/"n" + 4)`
= `lim_("n" -> oo) sum_("r" = 1)^"n" ("r"/"n"^2 + 4/"n")`
= `lim_("n" -> oo) [1/"n"^2 sum_("r" = 1)^"n" "r" + 4/"n" sum_("r" = 1)^"n" 1]`
= `lim_("n" -> oo) 1/"n"^2 [("n"("n" + 1))/2] + 4/"n" ("n")]`
= `lim_("n" -> oo) [1/2[1 + 1/"n"] + 4]`
= `1/2[1 + 0] + 4`
= `1/2 + 4`
= `9/2`
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