Advertisements
Advertisements
प्रश्न
Evaluate: `int_0^pi 1/(3 + 2sinx + cosx) "d"x`
उत्तर
Let I = `int_0^pi 1/(3 + 2sinx + cosx) "d"x`
Put `tan (x/2)` = t
∴ x = 2tan−1t
∴ dx = `(2"dt")/(1 + "t"^2)`, sin x = `(2"t")/(1 + "t"^2)` and cos x = `(1 - "t"^2)/(1 + "t"^2)`
When x = 0, t = 0 and when x = π, t = ∞
∴ I = `int_0^∞ 1/(3 + 2((2"t")/(1 + "t"^2)) + (1 - "t"^2)/(1 + "t"^2)) xx (2 "dt")/(1 + "t"^2)`
= `int_0^∞ (2 "dt")/(3 + 3"t"^2 + 4"t" + 1 - "t"^2)`
= `int_0^∞ (2 "dt")/(2"t"^2 + 4"t" + 4)`
= `int_0^∞ "dt"/("t"^2 + 2"t" + 2)`
= `int_0^∞ "dt"/("t"^2 + 2"t" + 1 + 1)`
= `int_0^∞ "dt"/(("t" + 1)^2 + 1^2)`
= `[tan^-1 ("t" + 1)]_0^∞`
= `tan^-1(1 + ∞) - tan^-1(1 + 0)`
= `tan^-1(∞) - tan^-1 (1)`
= `pi/2 - pi/4`
∴ I = = `pi/4`
संबंधित प्रश्न
Evaluate: `int_0^1 (x^2 - 2)/(x^2 + 1).dx`
Evaluate: `int_0^(pi/2) x sin x.dx`
`int_0^(x/4) sqrt(1 + sin 2x) "d"x` =
If `int_0^1 ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3`, then k is equal to ______.
`int_0^1 (x^2 - 2)/(x^2 + 1) "d"x` =
`int_0^4 1/sqrt(4x - x^2) "d"x` =
`int_0^(pi/2) log(tanx) "d"x` =
Evaluate: `int_0^1 1/(1 + x^2) "d"x`
Evaluate: `int_0^(pi/4) sec^2 x "d"x`
Evaluate: `int_1^2 x/(1 + x^2) "d"x`
Evaluate: `int_0^(pi/2) (sin2x)/(1 + sin^2x) "d"x`
Evaluate: `int_(pi/6)^(pi/3) sin^2 x "d"x`
Evaluate: `int_0^(pi/2) cos^3x "d"x`
Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`
Evaluate: `int_3^8 (11 - x)^2/(x^2 + (11 - x)^2) "d"x`
Evaluate: `int_(-1)^1 |5x - 3| "d"x`
Evaluate: `int_(-4)^2 1/(x^2 + 4x + 13) "d"x`
Evaluate: `int_0^1 x* tan^-1x "d"x`
Evaluate: `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2) "d"x`
Evaluate: `int_0^(pi/4) sec^4x "d"x`
Evaluate: `int_0^(pi/2) cos x/((1 + sinx)(2 + sinx)) "d"x`
Evaluate: `int_0^"a" 1/(x + sqrt("a"^2 - x^2)) "d"x`
Evaluate: `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x + 1) "d"x`
Evaluate: `int_0^1 (log(x + 1))/(x^2 + 1) "d"x`
Evaluate: `int_0^pi x*sinx*cos^2x* "d"x`
Evaluate: `int_0^(pi/4) log(1 + tanx) "d"x`
`int_0^(π/2) sin^6x cos^2x.dx` = ______.
Evaluate:
`int_(π/4)^(π/2) cot^2x dx`.
Evaluate:
`int_0^(π/2) sin^8x dx`
Evaluate `int_(π/6)^(π/3) cos^2x dx`
Evaluate:
`int_-4^5 |x + 3|dx`
The value of `int_2^(π/2) sin^3x dx` = ______.
Evaluate:
`int_0^(π/2) (sin 2x)/(1 + sin^4x)dx`
`int_0^1 x^2/(1 + x^2)dx` = ______.
Find the value of ‘a’ if `int_2^a (x + 1)dx = 7/2`
Evaluate:
`int_0^(π/2) sinx/(1 + cosx)^3 dx`
If `int_0^π f(sinx)dx = kint_0^π f(sinx)dx`, then find the value of k.
