Advertisements
Advertisements
प्रश्न
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Advertisements
उत्तर
Let I = `int 1/(3+2sinx + cosx) dx`
Put tan `x/2 = t` Then `dx = 2/(1+ t^2) dt`
`sinx = (2t)/(1+t^2) and cos x = (1- t^2)/(1+ t^2)`
`:. I = int (2dt"/" (1+t^2))/(3+2 ((2t)/(1+t^2))+((1-t^2)/(1+t^2)))`
`= 2int (dt"/"(1+t^2))/((3(1+t^2) + 4t + (1-t^2))/(1+t^2))`
= `2int (dt)/(2t^2 + 4t + 4) = int (dt)/((t+1)^2 + 1)`
`= tan^(-1) (t + 1) + c`
`= tan^(-1)[tan (x/2) + 1)] + c`
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Write a value of
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).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^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
`int (log x)/(log ex)^2` dx = _________
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int dx/(1 + e^-x)` = ______
`int (cos x)/(1 - sin x) "dx" =` ______.
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
Write `int cotx dx`.
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
