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 `intsqrtx/sqrt(a^3-x^3)dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : sin5x.cos8x
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate: `int "e"^sqrt"x"` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int cos sqrtx` dx = _____________
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int cot^2x "d"x`
`int cos^7 x "d"x`
`int sin^-1 x`dx = ?
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int (cos x)/(1 - sin x) "dx" =` ______.
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
