Advertisements
Advertisements
प्रश्न
Evaluate:
∫ (log x)2 dx
Advertisements
उत्तर
Let I = ∫ (log x)2 dx
I = ∫ (log x)2 . 1 dx
I = `(log x)^2 int 1. "dx" - int ["d"/"dx" (log x)^2 int 1. "dx"] "dx"`
I = `x(log x)^2 - int 2 log x. 1/cancelx. cancelx "dx"`
I = `x(log x)^2 - 2 int log x. 1 "dx"`
I = `x(log x)^2 - 2[log x int 1. "dx" - int {"d"/"dx" (log x) int 1. "dx"}]`dx
I = `x(log x)^2 - 2[(log x)x - int 1/cancelx. cancelx. "dx"]`
I = `x(log x)^2 - 2[xlog x - int 1. "dx"]`
I = x(log x)2 – 2(x log x – x) + c
∴ I = x(log x)2 – 2x log x + 2x + c
संबंधित प्रश्न
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int (sinx)/(1 + sin x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
`int 1/sqrt(x^2 - 8x - 20) "d"x`
∫ log x · (log x + 2) dx = ?
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
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 ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int1/(x+sqrt(x)) dx` = ______
`int(xe^x)/((1+x)^2) dx` = ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`inte^x sinx 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`
