Advertisements
Advertisements
Question
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Advertisements
Solution
Let I = `int (1)/(5 - 4x - 3x^2).dx`
5 – 4x –3x2 = `[5/3 - (x^2 + 4/3 x)]`
= `3[(5)/(3) - (x^2 + (4)/(3)x + (4)/(9)) + 4/9]`
= `3[(19)/(9) - (x^2 + (4x)/(3) + (4)/(9))]`
= `3[(sqrt(19)/3)^2 - (x + 2/3)^2]`
I = `int (1)/(3[(sqrt(19)/3)^2 - (x + 2/3)^2]).dx`
= `(1)/(3) (1)/(2(sqrt(19)/3))log |(sqrt(19)/(3) + (x + 2/3))/(sqrt(19)/(3) - (x + 2/3))| + c`
= `(1)/(2sqrt(19))log |(sqrt(19) + 2 + 3x)/(sqrt(19) - 2 - 3x)| + c`
= `(1)/(2sqrt(19))log |(3x + 2 + sqrt(19))/(-(3x + 2 - sqrt(19)))| + c`
= `(1)/(2sqrt(19))log |(3x + 2 + sqrt(19))/(3x + 2 - sqrt(19))| + c`. ...[∵ | – x |= x]
APPEARS IN
RELATED QUESTIONS
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : sin4x.cos3x
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 : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate `int 1/((2"x" + 3))` dx
`int cot^2x "d"x`
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
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 ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
`int secx/(secx - tanx)dx` equals ______.
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 the following.
`int1/(x^2+4x-5)dx`
Evaluate `int(5x^2-6x+3)/(2x-3)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).
