Advertisements
Advertisements
प्रश्न
Integrate the functions:
`1/(1 + cot x)`
उत्तर
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
संबंधित प्रश्न
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
The value of \[\int\frac{1}{x + x \log x} dx\] is
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : tan5x
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3)dx`
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: ∫ |x| dx if x < 0
Evaluate: `int "x" * "e"^"2x"` dx
Choose the correct alternative:
`int(1 - x)^(-2) dx` = ______.
`int1/(4 + 3cos^2x)dx` = ______
`int (f^'(x))/(f(x))dx` = ______ + c.
`int 1/(sinx.cos^2x)dx` = ______.
`int (logx)^2/x dx` = ______.
`int secx/(secx - tanx)dx` equals ______.
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
