Advertisements
Advertisements
Question
Integrate the functions:
`1/(1 + cot x)`
Advertisements
Solution
Let `I = int 1/ (1 + cot x) dx = int 1/ (1 + cos x/sinx) dx`
`= int sin/(sin x + cos x) dx`
`= 1/2 int (2 sin x)/ (sinx + cos x) dx`
`= 1/2 int ((sin x + cos x) - (cos x - sin x))/ ((sin x + cos x)) dx`
`= 1/2 int 1 dx - 1/2 int (cos x - sin x)/ (sin x + cos x) dx`
`= 1/2 x - 1/2 int (cos x - sin x)/ (sin x + cos x) dx + C_1`
`I = x/2 - 1/2 I_1 + C_1` ........(i)
Where, `I_1 = int (cos x - sin x)/ (sin x + cos x) dx`
Put sin x + cos x = t
⇒ (cos x - sin x) dx = dt
⇒ `I_1 = int dt/t = log |t| + C_2`
`= log |cos x + sin x| + C_2` ......(ii)
From (i) and (ii), we get
⇒ `I = 1/2 x - 1/2 log |cos x + sin x| + C`
APPEARS IN
RELATED QUESTIONS
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`(1+ log x)^2/x`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate `int (5"x" + 1)^(4/9)` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int cos sqrtx` dx = _____________
`int (log x)/(log ex)^2` dx = _________
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate `int1/(x(x - 1))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate:
`int sin^3x cos^3x dx`
