Advertisements
Advertisements
प्रश्न
Evaluate the definite integral:
`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`
उत्तर
Let `I = int_1^4 (|x - 1| + |x - 2| + |x - 3|) dx`
Define,
|x - 1| = x -1, when x - 1 ≥ 0, i.e., x ≥ 1
|x - 2| = x -2, when x - 2 ≥ 0, i.e., x ≥ 2
|x - 2| = - (x - 2), when x - 2 ≤ 0, i.e., x ≤ 2
|x - 3| = - (x - 3), when x - 3 ≤ 0, i.e., x ≤ 3
|x - 3| = (x - 3), when x - 3 ≥ 0, i.e, x ≥ 3
⇒ `I = int_1^4 (x - 1) dx - int_1^2 (x - 2) dx + int_2^4 (x - 2) dx - int_1^3 (x - 3) dx + int_3^4 (x - 3) dx`
`= [x^2/2 - x]_1^4 - [x^2/2 - 2x]_1^2 + [x^2/2 - 2x]_2^4 - [x^2/2 - 3x]_1^3 + [x^2/2 - 3x]_3^4`
`= [(16/2 - 1/2) - (4 - 1)] - [(4/2 - 1/2) - (4 - 2)] + [(16/2 - 1/2) - (8 - 4) - [(9/2 - 1/2) - (9 - 3)] + [(16/2 - 9/2) - (12 - 9)]`
`= [15/2 - 3/2 + 12/2 - 8/2 + 7/2] + [-3 + 2 - 4 + 6 - 3]`
`= [23/2] + [-2]`
`= 19/2`
APPEARS IN
संबंधित प्रश्न
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_1^4 (x^2 - x) dx`
Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Evaluate the definite integral:
`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Prove the following:
`int_0^1 xe^x dx = 1`
Evaluate `int_0^1 e^(2-3x) dx` as a limit of a sum.
`int dx/(e^x + e^(-x))` is equal to ______.
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.
(A) `1/2`
(B) `1/3`
(C) `1/4`
(D) `1/5`
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integrals as limit of sums:
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Evaluate:
`int (sin"x"+cos"x")/(sqrt(9+16sin2"x")) "dx"`
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/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
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→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` 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 ______.
