Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
Solution
Let f (x) = sin7 x.
sin x is an odd function
i.e. if h (x) = sin x
⇒ h (-x) = sin (-x)
= - sin (x) = -h (x)
⇒ odd power of sin x is odd
⇒ f (x) is an odd function of x.
⇒ `int_(-pi/2)^(pi/2) sin^7 x dx = 0` .... [∵ If f (x) is odd ⇒`int_-a^a` f (x) dx = 0]
APPEARS IN
RELATED QUESTIONS
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
`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_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
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 `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
`int_0^1|3x - 1|dx` equals ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` 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`.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate: `int_-1^1 x^17.cos^4x dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
