Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Advertisements
उत्तर
Let f (x) = cos5 x
Now we have
f (2π - x) = (cos (2π - x))5
= (cos x)5 = cos5 x = f (x)
⇒ `I = 2 int_0^pi cos^5 x dx`
`[∵ int_0^(2a) f (x) dx = 2 int_0^a f (x)dx, if (2a - x) = f(x) = 0, if (2a - x) = -f(x)]`
Again, we have
f (π - x) = (cos (π - x))5 = -cos5 x = - f(x)
⇒ `2 int_0^pi cos^5 x dx = 0`
Hence, `int_0^(2pi) cos^5 x dx `
`= 2 int_0^5 cos^5 x dx `
= 2 × 0
= 0
APPEARS IN
संबंधित प्रश्न
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
`int_0^1 "e"^(2x) "d"x` = ______
`int_1^2 1/(2x + 3) dx` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
`int_0^1 ((x^2 - 2)/(x^2 + 1))`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 ______.
`int_0^{pi/2} xsinx dx` = ______
`int_0^1 (1 - x)^5`dx = ______.
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
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 ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
`int_1^2 x logx dx`= ______
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following definite integral:
`int_1^3 log x dx`
