Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Advertisements
उत्तर
Let I = `int e^(5x) [(5x.log x + 1)/x].dx`
= `int e^(5x)[5log x + 1/x].dx`
Put 5x = t
∴ 5.dx = dt
∴ dx = `(1)/(5).dt`
Also, x = `t/(5)`
∴ I = `(1)/(5) int e^t [5 log (t/5) + 5/t].dt`
Let f(t) = `5log (t/5)`
= 5 log t – 5 log 5
∴ f'(t) = `d/dt [5log t - 5 log 5]`
= `(5)/t - 0`
= `(5)/t`
∴ I = `(1)/(5) int e^t [f(t) + f^'(t)].dt`
= `(1)/(5) e^t f(t) + c`
= `(1)/(5) e^t . 5log (t/5) + c`
= e5x log x + c.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Integrate the function in x sin 3x.
Integrate the function in x log x.
Integrate the function in x sin−1 x.
Integrate the function in x cos-1 x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
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 w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/("9x"^2 - 25)`
`int (sinx)/(1 + sin x) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
Find: `int e^x.sin2xdx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`int(1-x)^-2 dx` = ______
`int1/sqrt(x^2 - a^2) dx` = ______
`inte^(xloga).e^x dx` is ______
`int logx dx = x(1+logx)+c`
Evaluate:
`int (logx)^2 dx`
Evaluate `int tan^-1x dx`
Evaluate the following.
`intx^3 e^(x^2) 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 f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
