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
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin 3x.
Integrate the function in x log x.
Integrate the function in x tan-1 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in ex (sinx + cosx).
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int log(logx)/x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sinx)/(1 + sin x) "d"x`
`int 1/(4x + 5x^(-11)) "d"x`
`int(x + 1/x)^3 dx` = ______.
`int"e"^(4x - 3) "d"x` = ______ + c
`int "e"^x x/(x + 1)^2 "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int_0^1 x tan^-1 x dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
`int1/(x+sqrt(x)) dx` = ______
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`intcos^-1(sqrt(x))dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate `int tan^-1x dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate `int(1 + x + x^2/(2!))dx`.
