Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
Advertisements
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
Evaluate : `intlogx/(1+logx)^2dx`
Evaluate: `int_(-a)^asqrt((a-x)/(a+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)`
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate : ∫ log (1 + x2) dx
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
`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_0^4 1/(1 + sqrtx)`dx = ______.
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^1 x tan^-1x dx` = ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`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:
`int (dx)/(e^x + e^(-x))` is equal to ______.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a 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 `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k 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 ______.
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 ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
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`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
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_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
