Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
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
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
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 ______.
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_0^1 x tan^-1x dx` = ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` 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 ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
