Advertisements
Advertisements
Question
Evaluate the following : `int x^3.tan^-1x.dx`
Advertisements
Solution
Let I = `int x^3.tan^-1x.dx`
= `int (tan^-1 x).x^3dx`
= `(tan^-1x) int x^3.dx - int [{d/dx (tan^-1 x) int x^3.dx}].dx`
= `(tan^-1x) (x^4/4) - int (1/(1 + x^2))x^4/(4).dx`
= `x^4/(4) tan^-1x - (1)/(4) ((x^4 - 1) + 1)/(x^2 + 1)`
= `x^4/(4) tan^-1x - (1)/(4) int ((x^2 - 1)(x^2 + 1) + 1)/(x^2 + 1).dx`
= `x^4/(4) tan^-1x - (1)/(4) int [x^2 - 1 + 1/(x^2 + 1)].dx`
= `x^4/(4) tan^-1x - (1)/(4) int [int x^2.dx - int 1.dx + int 1/(x^2 + 1).dx]`
= `x^4/(4) tan^-1x - (1)/(4)[x^3/3 - x + tan^-1x] + c`
= `x^4/(4) tan^-1x - tan^-1 x/(4) - x^3/(12) - x/(4) + c`
= `(1)/(4) (tan^-1x) (x^4 - 1) - x/(12) (x^2 - 3) + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the function in x sin x.
Integrate the function in x log x.
`intx^2 e^(x^3) dx` equals:
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following: `int logx/x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
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)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : log (x2 + 1)
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int (sinx)/(1 + sin x) "d"x`
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int_0^1 x tan^-1 x dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int(1-x)^-2 dx` = ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int (logx)^2 dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
`∫ sin^(−1)` xdx is equal to ______.
