Advertisements
Advertisements
प्रश्न
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
विकल्प
`1/sqrt(3) tan^-1 ((tan(x/2))/sqrt(3)) + C`
`2/sqrt(3) tan^-1 ((tan(x/2))/sqrt(3)) + C`
`2/sqrt(3) tan^-1 ((tan(x/2))/(2sqrt(3))) + C`
`1/(2sqrt(3)) tan^-1 ((tan(x/2))/(2sqrt(3))) + C`
उत्तर
`int dx/(2 + cos x)` = `underlinebb(2/sqrt(3) tan^-1 ((tan(x/2))/sqrt(3)) + C)`.
(where C is a constant of integration)
Explanation:
Let I = `int dx/(2 + cos x)`
I = `int dx/(2 + (1 - tan^2 x/2)/(1 + tan^2 x/2))`
= `int ((1 + tan^2 x/2)dx)/(2 + 2tan^2 x/2 + 1 - tan^2 x/2)`
I = `int (sec^2 x/2 dx)/(3 + tan^2 x/2)`
Let `tan x/2` = t
`\implies` sec2x / 2dx = dt
∴ I = `int (2dt)/(3 + t^2)`
= `2int dt/((sqrt(3))^2 + t^2)`
I = `2 1/sqrt(3) tan^-1 (t/sqrt(3)) + C`
`\implies` I = `2/sqrt(3) tan^-1 ((tan(x/2))/sqrt(3)) + C`
