Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Advertisements
Solution
Let I = `int (1)/(3 + 2sin x - cosx)dx`
Put `tan(x/2)` = t
∴ x = 2 tan–1 t
∴ dx = `(2)/(1 + t^2)dt` and
sinx = `(2t)/(1 + t^2)' cosx = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(3 + 2((2t)/(1 + t^2)) - ((1 - t^2)/(1 + t^2))).(2dt)/(1 + t^2)`
= `int (1 + t^2)/(3(1 + t^2) + 4t - (1 - t^2)).(2dt)/(1 + t^2)`
= `2 int dt/(4t^2 + 4t + 2)`
= `2 int dt/(4t^2 + 4t + 1 + 1)`
= `2 int dt/((2t + 1)^2 + 1^2)`
= `(2)/(2)tan^-1((2t + 1)/1) + c`
= `tan^-1[2tan^-1(x/2) + 1] + c`.
APPEARS IN
RELATED QUESTIONS
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`(1+ log x)^2/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
The value of \[\int\frac{1}{x + x \log x} dx\] is
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
`int "dx"/(9"x"^2 + 1)= ______. `
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Evaluate `int 1/((2"x" + 3))` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int (cos2x)/(sin^2x) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
