Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
Advertisements
उत्तर
`int_2^8 abs (x - 5) dx`
Define,
`abs(x - 5) = {(-(x - 5), if x - 5 < 0, or x< 5),(x - 5, if x - 5 >= 0, or x >=5):}`
`= int_2^5 abs (x - 5) dx + int_2^8 abs (x - 5) dx`
`= - int_2^5 (x - 5) dx + int_2^8 (x - 5) dx`
`= - [x^2/2 - 5x]_2^5 + [x^2/2 - 5x]_5^8`
`= - [25/2 - 25 - 4/2 + 10]`
`= [64/2 - 4 - 25/2 + 25]`
`= - [(-9)/2] + [9/2]`
`= 9/2 + 9/2`
= 9
APPEARS IN
संबंधित प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_0^1 "e"^(2x) "d"x` = ______
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_0^{pi/2} log(tanx)dx` = ______
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0
⇒ `1/4 (square - square)` = 0
⇒ b4 – `square` = 0
⇒ (b2 – a2)(`square` + `square`) = 0
⇒ b2 – `square` = 0 as a2 + b2 ≠ 0
⇒ b = ± `square`
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
Evaluate the following definite integral:
`int_4^9 1/sqrt"x" "dx"`
Solve the following.
`int_1^3 x^2 logx dx`
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate: `int_-1^1 x^17.cos^4x dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
