Advertisements
Advertisements
Question
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Advertisements
Solution
Let I = `int ["cosec"(logx)][1 - cot(logx)] "d"x`
Put logex = t
∴ x = et
∴ dx = `"e"^"t"*"dt"`
∴ I = `int "cosec" "t"(1 - cot "t") "e"^"t" "dt"`
= `int "e"^"t" ("cosec" "t" - "cosec" "t"*cot "t") "dt"`
Put f(t) = cosec t
∴ f'(t) = −cosec t.cot t
∴ I = `int"e"^"t" ["f"("t") + "f'"("t")] "dt"`
= et ⋅ f(t) + c = et cosec t + c
∴ I = `x "cosec" (logx) + "c"`
RELATED QUESTIONS
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Integrate the function in x sin x.
Integrate the function in x2 log x.
Integrate the function in (sin-1x)2.
Integrate the function in (x2 + 1) log x.
Integrate the function in `(xe^x)/(1+x)^2`.
`int e^x sec x (1 + tan x) dx` equals:
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int 1/(4x + 5x^(-11)) "d"x`
`int sin4x cos3x "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(4x^2 - 1) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
Find: `int e^x.sin2xdx`
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
`int logx dx = x(1+logx)+c`
`int(xe^x)/((1+x)^2) dx` = ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
Under the LIATE rule, which type of function has priority \(L\)?
