Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
Advertisements
उत्तर
Let `I = int_-5^5 abs (x + 2) dx`
Define,
`abs (x + 2) = {(-(x + 2), if x + 2 < 0, or x< - 2),(x + 2, if x +2 >= 0, or x >=-2):}`
∵ `I = - int_-5^-2 (x + 2) dx + int_-2^5 (x + 2) dx`
`= -[(x + 2)^2/2]_-5^-2 + [(x + 2)^2/2]_-2^5`
`= [((-2 + 2)^2/2 - (-5 + 2)^2/2)] + [(5 + 2)^2/2 - (-2 + 2)^2/2]`
`= -1/2 [-9] + 1/2 [49 - 0]`
`= 9/2 + 49/2`
`= 58/2`
= 29
APPEARS IN
संबंधित प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_0^2 e^x dx` = ______.
Evaluate `int_0^1 x(1 - x)^5 "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_0^4 1/(1 + sqrtx)`dx = ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^1 x tan^-1x dx` = ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_0^1 "e"^(5logx) "d"x` = ______.
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
