Advertisements
Advertisements
Question
Integrate the functions:
`1/(1 + cot x)`
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
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
tan2(2x – 3)
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following integrals : `int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
If f'(x) = `x + 1/x`, then f(x) is ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate:
`int sqrt((a - x)/x) dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate:
`intsqrt(sec x/2 - 1)dx`
