Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
Advertisements
उत्तर
`int_0^1 (1 - x) [1 - (1 - x)^n] dx ...[because int_0^a f(x) dx = int_0^a f(a - x) dx]`
Hence, `I = int_0^1 (1 - x).x^n dx`
`I = int_0^1 (x^n - x^(n + 1)) dx`
`= ([x^(n + 1)]_0^1)/(n + 1) - ([n^(n + 2)]_0^1)/(n + 2)`
`= 1/(n + 2) - 1/(n + 2)`
`= (n + 2 - n - 1)/((n + 1)(n + 2))`
`= 1/((n + 1)(n + 2))`
APPEARS IN
संबंधित प्रश्न
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
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 (x dx)/(1+ sin x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
Evaluate : `int _0^(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.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - 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_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_2^3 x/(x^2 - 1)` dx = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
Evaluate: `int_(-1)^3 |x^3 - x|dx`
`int_a^b f(x)dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
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^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate: `int_-1^1 x^17.cos^4x dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
