Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Advertisements
Solution
Let I = `int(1)/(4x + 5x^-11).dx`
= `int x_11/(x^11(4x + 5x^-11)).dx`
= `int x^11/(4x^12 + 5).dx`
= `(1)/(48) int(48x^11)/(4x^12 + 5).dx`
= `(1)/(48) int(d/dx(4x^12 + 5))/(4x^12 + 5).dx`
= `(1)/(48)log|4x^12 + 5| + c ...[∵ int (f'(x))/f(x) dx = log|f(x)| + c]`
APPEARS IN
RELATED QUESTIONS
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`(1+ log x)^2/x`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
`int cos sqrtx` dx = _____________
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
If f'(x) = 4x3- 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate:
`int sin^3x cos^3x dx`
Which substitution is appropriate for \[\sqrt{\frac{x}{x-a}}\], \[\sqrt{\frac{x-a}{x}}\], \[\sqrt{x(x-\mathrm{a})}\], or \[\frac{1}{\sqrt{x(x-\mathrm{a})}}\]?
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
How should a substitution be chosen?
