Advertisements
Advertisements
Question
Integrate the following w.r.t.x : log (log x)+(log x)–2
Advertisements
Solution
Let I = `int [log (logx) + (logx)^-2]*dx`
= `int [log(logx) + 1/(log x)^2]*dx`
Put log x = t
∴ x = et
∴ x = et·dt
∴ I = `int (log t + 1/t^2)e^t*dt`
= `int e^t (log t + 1/t - 1/t + 1/t^2)*dt`
= `int [e^t (log t 1/t) + e^t (-1/t + 1/t^2)]*dt`
= `inte^t (log t + 1/t)*dt - int e^t (1/t - 1/t^2)*dt`
= I1 – I2
In I1, Put f(t) = log t. Then f'(t) = `(1/t)`
∴ I1 = `int e^t [f(t) + f'(t)]*dt`
= `e^t f(t)`
= `e^t log t`
In I2, Put g(t) = `(1/t)`. Then g'(t) = `-(1/t^2)`
∴ I2 = `int e^t ["g"(t) + "g"'(t)]*dt`
= `e^t "g" (t)`
= `e^t*(1/t)`
∴ I = `e^t log t - e^t/t + c`
= `xlog (logx) - x/logx + c`.
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x log 2x.
Integrate the function in x2 log x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`intx^2 e^(x^3) 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`
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*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 w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
`int 1/x "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
`int log x * [log ("e"x)]^-2` dx = ?
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
`int 1/sqrt(x^2 - a^2)dx` = ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
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:
`int1/(x^2 + 25)dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
Which pair is listed as Trigonometric in the LIATE rule?
Which pair is listed as Exponential in the LIATE rule?
Using \[t=1-x^2,\] what is \[\int \frac{x\,dx}{\sqrt{1-x^2}}?\]
Evaluate \[\int e^x\sin x\,dx.\]
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
Repeated parts may be needed for which pair of integrals?
