Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Advertisements
Solution
Let I = `int "cosec" (log x)[1 - cot (log x)].dx`
Put log x = t
∴ et
∴ dx = et .dt
∴ I = `int "cosec" t (1 - cot t).e^t dt`
= `int e^t ["cosec" t - "cosec" t cot t].dt`
= `int e^t ["cosec" t + d/dt ("cosec" t)].dt`
= `e^t "cosec" t + c ...[∵ int e^t [f(t) + f'(t)].dt = e^t f(t) + c]`
= x . cosec (log x) + c.
APPEARS IN
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 log x.
Integrate the function in (sin-1x)2.
Integrate the function in tan-1 x.
Integrate the function in e2x sin x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
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 x^3.logx.dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "x"^2 *"e"^"3x"`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/("9x"^2 - 25)`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int ("d"x)/(x - x^2)` = ______
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
Evaluate `int 1/(x log x) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int 1/sqrt(x^2 - a^2)dx` = ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`intsqrt(1+x) dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`inte^x sinx dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
