Advertisements
Advertisements
प्रश्न
Find `int_ (log "x")^2 d"x"`
उत्तर
Let `I = int (log "x")^2 d"x"`
⇒ `I = int_ 1·(log "x")^2 d"x"`
⇒ `I = "x"·(log "x")^2 - int_ (2"x" log"x")/"x" d"x"`
⇒ `I = "x"·(log "x")^2 - I_1 + c_1` .....(i)
`I_1 = int_ 2·log "x"d"x"`
⇒ `I_1 = 2"x"· log"x"- 2 int_ "x"/"x" d"x"`
⇒ `I_1 = 2"x"·log "x" - 2"x" + c_2` .....(ii)
From (i) and (ii), we get
`I = "x"·(log "x")^2 - 2"x"·log "x"+ 2"x" + c_1 - c_2`
`I = "x"·(log "x")^2 - 2"x"·log "x"+ 2"x" + C` ...(where C = C1 - C2)
APPEARS IN
संबंधित प्रश्न
Find the integrals of the function:
sin2 (2x + 5)
Find the integrals of the function:
sin 3x cos 4x
Find the integrals of the function:
cos 2x cos 4x cos 6x
Find the integrals of the function:
sin3 x cos3 x
Find the integrals of the function:
sin x sin 2x sin 3x
Find the integrals of the function:
sin4 x
Find the integrals of the function:
cos4 2x
Find the integrals of the function:
`(cos 2x - cos 2 alpha)/(cos x - cos alpha)`
Find the integrals of the function:
`(sin^3 x + cos^3 x)/(sin^2x cos^2 x)`
Find the integrals of the function:
`(cos 2x+ 2sin^2x)/(cos^2 x)`
Find the integrals of the function:
`1/(sin xcos^3 x)`
`int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx` is equal to ______.
Find `int dx/(x^2 + 4x + 8)`
Evaluate `int_0^pi (x sin x)/(1 + cos^2 x) dx`
Find `int_ (sin2"x")/((sin^2 "x"+1)(sin^2"x"+3))d"x"`
Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration.
Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.
Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`
`int "e"^x (cosx - sinx)"d"x` is equal to ______.
`int "dx"/(sin^2x cos^2x)` is equal to ______.
Evaluate the following:
`int ((1 + cosx))/(x + sinx) "d"x`
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
Evaluate the following:
`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
`int (cos^2x)/(sin x + cos x)^2 dx` is equal to
