Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
Evaluate : `intsec^nxtanxdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_"a"^"b" "f"(x) "d"x` = ______
`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` =
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_0^{pi/2} cos^2x dx` = ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
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_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
`int_0^1 1/(2x + 5) dx` = ______.
`int_0^1|3x - 1|dx` equals ______.
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 ______.
`int_0^(π/4) x. sec^2 x dx` = ______.
Evaluate `int_-1^1 |x^4 - x|dx`.
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`.
`int_1^2 x logx dx`= ______
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is ______.
