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
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_0^1 "e"^(2x) "d"x` = ______
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^4 1/(1 + sqrtx)`dx = ______.
`int_-9^9 x^3/(4 - x^2)` dx = ______
If `int_0^"a" sqrt("a - x"/x) "dx" = "K"/2`, then K = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_0^1 "e"^(5logx) "d"x` = ______.
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
`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"`
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
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 ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Solve the following.
`int_1^3 x^2 logx dx`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Evaluate:
`int_0^6 |x + 3|dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
