Advertisements
Advertisements
Question
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Advertisements
Solution
Let I = `int 1/sqrt((x - 3)(x + 2)).dx`
= `int 1/sqrt(x^2 - x - 6).dx`
= `int 1/sqrt((x^2 - x + 1/4) - 1/4 - 6).dx`
= `int 1/sqrt((x - 1/2)^2 - (5/2)^2).dx`
= `log|(x - 1/2) + sqrt((x - 1/2)^2 - (5/2)^2)| + c`
= `log|(x - 1/2) + sqrt(x^2 - x - 6)| + c`.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
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 (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Evaluate: `int log ("x"^2 + "x")` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
Which substitution is appropriate for \[\sqrt{\frac{x}{x-a}}\], \[\sqrt{\frac{x-a}{x}}\], \[\sqrt{x(x-\mathrm{a})}\], or \[\frac{1}{\sqrt{x(x-\mathrm{a})}}\]?
What is the value of \[\int\sin^3x\cos^2x\,dx\]?
What is \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
When applying substitution, what must always be rewritten in terms of the new variable?
