Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Advertisements
उत्तर
Let I = `int [1/(log "x") - 1/(log "x")^2]` dx
Put log x = t
∴ x = et
∴ dx = et dt
∴ I = `int "e"^"t" [1/"t" - 1/"t"^2]` dt
Put f(t) = `1/"t"`
∴ f '(t) = `(-1)/"t"^2`
∴ I = `int "e"^"t" ["f"("t") + "f" '("x")]` dt
`= "e"^"t" "f"("t")` + c
∴ I = `"e"^"t" (1/"t") + "c" = "x"/(log "x")` + c
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin x.
Integrate the function in `x^2e^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^3.tan^-1x.dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
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_0^1 x tan^-1 x dx` = ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
