Advertisements
Advertisements
प्रश्न
`int logx/(1 + logx)^2 "d"x`
उत्तर
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
संबंधित प्रश्न
Integrate the function in x2 log x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in (x2 + 1) log x.
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 the following:
`int x.sin 2x. cos 5x.dx`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int 1/(4x + 5x^(-11)) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int(1-x)^-2 dx` = ______
`int1/sqrt(x^2 - a^2) dx` = ______
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
