Advertisements
Advertisements
Question
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Advertisements
Solution
Let I = `int "dx"/(4"x"^2 - 1)`
`= 1/4 int "dx"/("x"^2 - 1/4)`
`= 1/4 int "dx"/("x"^2 - (1/2)^2)`
`= 1/4 xx 1/(2 xx 1/2) log |("x" - 1/2)/("x" + 1/2)|` + c
∴ I = `1/4` log `|("2x" - 1)/("2x" + 1)|` + c
Alternate Method:
Let I = `int "dx"/(4"x"^2 - 1) = int "dx"/((2"x"^2) - (1)^2)`
`= 1/(2 xx 1) xx 1/2 log |("2x" - 1)/("2x" + 1)|` + c
∴ I = `1/4` log `|("2x" - 1)/("2x" + 1)|` + c
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
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
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate `int "x - 1"/sqrt("x + 4")` dx
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int (1)/(x(x - 1))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)
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
What is integration by substitution?
