Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Advertisements
Solution
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
RELATED QUESTIONS
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total revenue R is increasing.
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
`int_0^2 e^x dx` = ______.
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_0^1 (1 - x)^5`dx = ______.
`int_0^{pi/2} cos^2x dx` = ______
`int_0^1 x tan^-1x dx` = ______
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_0^pi x*sin x*cos^4x "d"x` = ______.
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int (dx)/(e^x + e^(-x))` is equal to ______.
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
If \[f(-x)=-f(x)\], what is \[\int_{-a}^{a} f(x)\,dx\]?
