Advertisements
Advertisements
प्रश्न
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Advertisements
उत्तर
We know that `int_"a"^"b" "f"(x) "d"x = lim_("n" -> oo) "h" sum_("r" = 0)^("n" - 1) "f"("a" + "rh")`
For I = `int_0^2 (x^2 + 3) "d"x`
We have a = 0 and b = 2
I = `int_00^2 (x^2 + 3) "d"x`
Here, a = 0, b = 2 and h = `("b" - "a")/"n" = (2 - 0)/"n" = 2/"n"`
⇒ nh = 2
And f(x) = `(x^2 + 3)`
∴ I = `int_0^2 (x^2 + 3)"d"x = lim_("h" -> 0) "h" sum_("r" = 0)^("n" - 1) "f"("a" + "rh")`
= `lim_("h" -> 0) "h" sum_("r" = 0)^("n" - 1) "f"("rh")`
= `lim_("h" -> 0) "h" sum_("r" = 0)^("n" - 1) (3 + "r"^2"h"^2)`
= `lim_("h" -> 0) "h"[3"n" + "h"^2 ((("n" - 1)("n" - 1 + 1)(2"n" - 2 + 1))/6)]`
= `lim_("h" -> 0) "h"[3"n" + "h"^2 ((("n"^2 - "n")(2"n" - 1))/6)]`
= `lim_("h" -> 0) "h" [3"n" + "h"^2/6 (2"n"^3 - 3"n"^2 + "n")]`
= `lim_("h" -> 0) [3"nh" + (2"n"^3"h"^3 - 3"n"^2"h"^2 * "h" + "nh" * "h"^2)/6]`
= `lim_("h" -> 0) [3.2 + (2.2^3 - 3.2^2 * "h" + 2 * "h"^2)/6]`
= `6 + 16/6`
= `26/3`
APPEARS IN
संबंधित प्रश्न
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
If f" = C, C ≠ 0, where C is a constant, then the value of `lim_(x -> 0) (f(x) - 2f (2x) + 3f (3x))/x^2` is
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.
`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to ______.
