Advertisements
Advertisements
Question
`int logx/(1 + logx)^2 "d"x`
Advertisements
Solution
Let I = `int logx/(1 + logx)^2 "d"x`
Put log x = t
∴ x = et
∴ dx = et dt
∴ I = `int "t"/(1 + "t")^2 "e"^"t" "dt"`
= `int "e"^"t" [(("t" + 1) - 1)/(1 + "t")^2] "dt"`
= `int "e"^"t" [("t" + 1)/(1 + "t")^2 - 1/(1 + "t")^2] "dt"`
= `int "e"^"t" [1/(1 + "t") - 1/(1 + "t")^2] "dt"`
Put f(t) = `1/(1 + "t")`
∴ f'(t) = `(-1)/(1 + "t")^2`
∴ I = `int "e"^"t" ["f"("t") + "f'"("t")] "dt"`
= et f(t) + c
= `"e"^"t"* 1/(1 + "t") + "c"`
∴ I = `x/(1 + logx) + "c"`
APPEARS IN
RELATED QUESTIONS
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in (x2 + 1) log x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x^3.logx.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Evaluate the following.
`int x^3 e^(x^2)`dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int 1/sqrt(x^2 - 9) dx` = ______.
Find: `int e^x.sin2xdx`
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
Evaluate \[\int e^x\sin x\,dx.\]
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
