Advertisements
Advertisements
Question
Integrate the functions:
`(1+ log x)^2/x`
Solution
Let `I = int (1 + log x)^2/x dx`
Put 1 + log x = t
`1/x dt = dt`
Hence, `I = int t^2` dt
`= t^3/3 + C`
`= 1/3 (1 + log x)^3 + C`
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(4"x"^2 - 20"x" + 17)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3)dx`
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int 1/(xsin^2(logx)) "d"x`
`int cot^2x "d"x`
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"`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int dx/(1 + e^-x)` = ______
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`