Advertisements
Advertisements
Question
Evaluate the definite integral:
`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) 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_(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_0^1 dx/(sqrt(1+x) - sqrtx)`
Prove the following:
`int_1^3 dx/(x^2(x +1)) = 2/3 + log 2/3`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Evaluate `int_0^1 e^(2-3x) dx` as a limit of a sum.
Evaluate the following integral:
Evaluate the following integrals as limit of sums:
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
