Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Advertisements
Solution
Let I = `int cosx/sin(x - a).dx`
= `int cos[(x - a) + a]/sin(x - a).dx`
= `int[cos(x - a)cos a - sin(x - a)sin a)/sin(x - a).dx`
= `int [(cos(x - a)cos a)/sin(x - a) - (sin(x - a)sina)/sin(x - a)].dx`
= `cos a int cot (x - a)dx - sin a int 1 dx`
= cos a log |sin(x – a)| – x sin a + c.
APPEARS IN
RELATED QUESTIONS
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Evaluate the following : `int (logx)2.dx`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` 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 = ?
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int 1/(xsin^2(logx)) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int(1 - x)^(-2) dx` = ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int cos^3x dx` = ______.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
