Advertisements
Advertisements
Question
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Solution
Let `I = ((x - 3) e^x)/(x - 1)^3 dx`
`= int (e^x (x - 1 - 2))/(x - 1)^3 dx`
`= int e^x [1/((x - 1)^2) - 2/((x - 1)^3)] dx`
On substituting `e^x . 1/((x - 1)^2) = t`
`[e^x - 2 (x - 1)^-3 + 1/((x - 1)^2). e^x] dx = dt`
or `e^x [1/((x - 1)^2) - 2/(x - 1)^3] dx = dt`
Hence, `I = int 1. dt = t + C`
`= e^x/((x - 1)^2) + C`
APPEARS IN
RELATED QUESTIONS
Integrate the function in x log x.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in e2x sin x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
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: `(1 + log x)^2/x`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int 1/x "d"x` = ______ + c
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(x log x) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int 1/sqrt(x^2 - a^2)dx` = ______.
Evaluate:
`int e^(logcosx)dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3 e^(x^2)dx`
