Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int "x"^2 "e"^"3x"`dx
उत्तर
Let I = `int "x"^2 "e"^"3x"`dx
`= "x"^2 int "e"^"3x" "dx" - int["d"/"dx" ("x"^2) int "e"^"3x" "dx"]` dx
`= "x"^2 * ("e"^"3x"/3) - int 2"x" * "e"^"3x"/3` dx
`= ("x"^2)/3 "e"^"3x" - 2/3 int "x" * "e"^"3x"` dx
`= ("x"^2)/3 "e"^"3x" - 2/3 ["x" int "e"^"3x" "dx" - int ("d"/"dx" ("x") int "e"^"3x" "dx") "dx"]`
`= ("x"^2 * "e"^"3x")/3 - 2/3 ["x" * "e"^"3x"/3 - int 1 * "e"^"3x"/3 "dx"]`
`= ("x"^2 * "e"^"3x")/3 - 2/3 [1/3 "xe"^"3x" - 1/3 int "e"^"3x" "dx"]`
`= ("x"^2 * "e"^"3x")/3 - 2/3 [1/3 "xe"^"3x" - 1/3 * "e"^"3x"/3]` + c
∴ I = `1/3 "x"^2 * "e"^"3x" - 2/9 "xe"^"3x" + 2/27 "e"^"3x" + "c"`
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
Integrate : sec3 x w. r. t. x.
Integrate the function in x log 2x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in ex (sinx + cosx).
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : sec4x cosec2x
`int 1/x "d"x` = ______ + c
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`intsqrt(1+x) dx` = ______
`int1/(x+sqrt(x)) dx` = ______
Evaluate:
`int e^(logcosx)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
