Advertisements
Advertisements
Question
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Write a value of
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
`int logx/(log ex)^2*dx` = ______.
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
`int (logx)^2/x dx` = ______.
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate `int (1)/(x(x - 1))dx`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
