Advertisements
Advertisements
प्रश्न
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Advertisements
उत्तर
We have I = `int_0^(pi/2) (sin^2x)/(sinx + cosx) "d"x`
= `int_0^(pi/2) (sin^2(pi/2 - x))/(sin(pi/2 - x) + cos(pi/2 - x)) "d"x` ....(By P4)
⇒ I = `int_0^(pi/2) (cos^2x)/(sinx + cosx) "d"x`
Thus, we get 2I = `1/sqrt(2) int_0^(pi/2) ("d"x)/(cos(x - pi/4))`
= `1/sqrt(2) int_0^(pi/2) sec(x - pi/2) "d"x`
= `1/sqrt(2) [log(sec(x - pi/4) + tan(x - pi/4))]_0^(pi/2)`
= `1/sqrt(2)[log(sec pi/4 + tan pi/4) - log sec(- pi/4) + tan(- pi/4)]`
= `1/sqrt(2) [log(sqrt(2) + 1) - log(sqrt(2) - 1)]`
= `1/sqrt(2) log|(sqrt(2) + 1)/(sqrt(2) - 1)|`
= `1/sqrt(2) log((sqrt(2) - 1)^2/1)`
= `2/sqrt(2) log(sqrt(2) + 1)`
Hence I = `1/sqrt(2) log(sqrt(2) + 1)`.
APPEARS IN
संबंधित प्रश्न
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
Evaluate : `intlogx/(1+logx)^2dx`
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_0^1 "e"^(2x) "d"x` = ______
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_0^1 x tan^-1x dx` = ______
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
The value of `int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2)) dx` is
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
Evaluate `int_-1^1 |x^4 - x|dx`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
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 integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
