Advertisements
Advertisements
प्रश्न
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
विकल्प
True
False
Advertisements
उत्तर
True
Explanation:
Let I = ∫ x (xx)x (2 log x + 1) dx
Put `("x"^"x")^"x"` = t
Taking logarithm of both sides, we get
log `("x"^"x")^"x"` = log t
∴ `"x"^2 * log "x" = log "t"`
Differentiating w.r.t. x, we get
`"x"^2 * 1/"x" + (log "x") * "2x" = 1/"t" * "dt"/"dx"`
∴ `("x" + 2"x" log "x") "dx" = 1/"t" * "dt"`
∴ x(1 + 2 log x) dx = `1/"t" * "dt"`
∴ I = `int "t" * 1/"t" * "dt" = int 1 * "dt" = "t" + "c" = ("x"^"x")^"x"` + c
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Write a value of
Write a value of
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate `int (5"x" + 1)^(4/9)` dx
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate:
`int sin^2(x/2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
