Advertisements
Advertisements
प्रश्न
Evaluate the following integrals using properties of integration:
`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x)) "d"x`
उत्तर
Let I = `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x)) "d"x` .......(1)
`int_"a"^"b" f(x) "d"x = int_"a"^"b" f("a" + "b" - x) "d"x`
I = `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan (pi/8 + (3pi)/8 - x))) "d"x`
= `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan(pi/2 - x))) "d"x`
= `int_(pi/8)^((3pi)/8) 1/(1 + sqrt(cot x)) "d"x`
= `int_(pi/8)^((3pi)/8) 1/(1 + 1/sqrt(tan x)) "d"x`
I = `int_(pi/8)^((3pi)/8) sqrt(tanx)/(1 + sqrt(tan x)) "d"x` ........(2)
Add (1) + (2)
2I = `int_(pi/8)^((3pi)/8) (1 + sqrt(tan x))/(1 + sqrt(tan x)) "d"x`
= `int_(pi/8)^((3pi)/8) "d"x`
= `[x]_(pi/8)^((3pi)/8)`
= `(3pi)/8- pi/8`
2I = `(2pi)/8 = (pi/4)`
I = `pi/8`
`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x)) "d"x = pi/8`
APPEARS IN
संबंधित प्रश्न
Evaluate the following definite integrals:
`int_3^4 (d"x)/(x^2 - 4)`
Evaluate the following definite integrals:
`int_0^1 sqrt((1 - x)/(1 + x)) "d"x`
Evaluate the following definite integrals:
`int_0^(pi/2) "e"^x((1 + sin x)/(1 + cos x))"d"x`
Evaluate the following definite integrals:
`int_0^1 (1 - x^2)/(1 + x^2)^2 "d"x`
Evaluate the following integrals using properties of integration:
`int_(- pi/4)^(pi/4) sin^2x "d"x`
Evaluate the following integrals using properties of integration:
`int_0^(2pi) x log((3 + cosx)/(3 - cosx)) "d"x`
Evaluate the following integrals using properties of integration:
`int_0^pi sin^4 x cos^3 x "d"x`
Evaluate the following integrals using properties of integration:
`int_0^(sin^2x) sin^-1 sqrt("t") "dt" + int_0^(cos^2x) cos^-1 sqrt("t") "dt"`
Evaluate the following integrals using properties of integration:
`int_0^1 (log(1 + x))/(1 + x^2) "d"x`
Evaluate the following integrals using properties of integration:
`int_0^pi(xsinx)/(1 + sinx) "'d"x`
Choose the correct alternative:
The value of `int_(-4)^4 [tan^-1 ((x^2)/(x^4 + 1)) + tan^-1 ((x^4 + 1)/x^2)] "d"x` is
Choose the correct alternative:
The value of `int_(- pi/4)^(pi/4) ((2x^7 - 3x^5 + 7x^3 - x + 1)/(cos^2x)) "d"x` is
Choose the correct alternative:
The value of `int_0^pi ("d"x)/(1 + 5^(cosx))` is
