Advertisements
Advertisements
Question
Integrate the functions:
`1/(1 - tan x)`
Advertisements
Solution
Let `I = int 1/ (1 - tan x)dx = int 1/ (1 - sin x/ cos x) dx`
`= int cos x/ (cos x - sin x) dx = 1/2 int (2 cos x)/ (cos x - sin x) dx`
`1/2 int ((cos x - sin x) - (-sin x - cos x))/(cos x - sin x)`
`1/2 int 1 dx - 1/2 int (-sin x - cos x)/ (cos x - sin x) dx`
`x/2 - 1/2 int (-sin x - cos x)/ (cos x - sin x) dx + C_1`
`I = x/2 - 1/2 I_1 + C_1` ....(i)
Where, `I_1 = int (-sinx - cos x)/(cos x - sin x) dx`
Put cos x - sin x = t
⇒ (-sin x - cos x) dx = dt
`I_1 = int dt/t = log |t| + C_2`
= log | cos x - sin x| + C2 ...(ii)
From (i) and (ii), we get
⇒ `I = x/2 - 1/2 log |cos x - sin x| + C`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Write a value of
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate `int (3"x"^2 - 5)^2` dx
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: `int 1/(sqrt("x") + "x")` dx
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int(1 - x)^(-2) dx` = ______.
`int x^3"e"^(x^2) "d"x`
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int 1/(sinx.cos^2x)dx` = ______.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
`int x^3 e^(x^2) dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
