Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Advertisements
Solution
Let I = `int(x^n - 1)/sqrt(1 + 4x^n).dx`
Put xn = t
∴ nxn–1 dx = dt
∴ xn–1 dx = `dt/n`
∴ I = `int (1)/sqrt(1 + 4t).dt/n`
= `(1)/nint(1 + 4t)^(-1/2)dt`
= `1/n.((1 + 4t)^(1/2))/(1/2) xx (1)/(4) + c`
= `(1)/(2n).sqrt(1 + 4x^n) + c`.
APPEARS IN
RELATED QUESTIONS
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
`int logx/x "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"`
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate `int1/(x(x - 1))dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int(1+x+x^2/(2!))dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
