Advertisements
Advertisements
Question
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Advertisements
Solution
Let `I = int ((x + 1) (x + log x)^2)/x dx`
`= int (x + log x)^2 (1 + 1/x) dx`
Put x + log x = t
⇒ `(1 + 1/x) dx = dt`
∴ `I = int t^2 dt = t^3/3 + C`
`= 1/3 (x + log x)^3 + C`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`(1+ log x)^2/x`
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
`int cos sqrtx` dx = _____________
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
`int 1/(sinx.cos^2x)dx` = ______.
`int cos^3x dx` = ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Evaluate the following
`int1/(x^2 +4x-5)dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate:
`int(cos 2x)/sinx dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate `int (1 + x + x^2/(2!)) dx`
