Evaluate the following: d∫x2dxx4-x2-12 - Mathematics

Advertisements
Advertisements
Sum

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Advertisements

Solution

Let I = `int (x^2"d"x)/(x^4 - x^2 - 12)`

= `int x^2/(x^4 - 4x^2 + 3x^2 - 12) "d"x`

= `int x^2/(x^2(x^2 - 4) + 3(x^2 - 4)) "d"x`

= `int x^2/((x^2 - 4)(x^2 + 3)) "d"x`

Put x2 = t for the purpose of partial fraction.

We get `"t"/(("t" - 4)("t" + 3))`

Let `"t"/(("t" - 4)("t" + 3)) = "A"/("t" - 4) + "B"/("t" + 3)` .....[where A and B are arbitrary constants]

`"t"/(("t" - 4)("t" + 3)) = ("A"("t" + 3) + "B"("t" - 4))/(("t" - 4)("t" + 3))`

⇒ t = At + 3A + Bt – 4B

Comparing the like terms, we get

A + B = 1 and 3A – 4B = 0

⇒ 3A = 4B

∴ A = `4/3 "B"`

Now `4/3 "B" + "B"` = 1

`7/3 "B"` = 1

∴ B = `3/7` and A = `4/3 xx 3/7 = 4/7`

So, A = `4/7` and B = `3/7`

∴ `int x^2/((x^2 - 4)(x^2 + 3)) "d"x`

= `4/7 int 1/(x^2 - 4)  "d"x + 3/7 int 1/(x^2 + 3)  "d"x`

= `4/7 int 1/(x^2 - (2)^2) "d"x + 3/7 int 1/(x^2 + (sqrt(3)^2)  "d"x`

= `4/7 xx 1/(2 xx 2) log|(x - 2)/(x + 2)| + 3/7 xx 1/sqrt(3) tan^-1  x/sqrt(3)`

= `1/7 log |(x - 2)/(x + 2)| + sqrt(3)/7 tan^-1 x/sqrt(3) + "C"`

Hence, I = `1/7 log |(x - 2)/(x + 2)| + sqrt(3)/7 tan^-1  x/sqrt(3) + "C"`.

  Is there an error in this question or solution?
Chapter 7: Integrals - Exercise [Page 165]

APPEARS IN

NCERT Exemplar Mathematics Class 12
Chapter 7 Integrals
Exercise | Q 35 | Page 165

RELATED QUESTIONS

Find: `I=intdx/(sinx+sin2x)`


Evaluate: `∫8/((x+2)(x^2+4))dx` 


Integrate the rational function:

`1/(x^2 - 9)`


Integrate the rational function:

`(1 - x^2)/(x(1-2x))`


Integrate the rational function:

`x/((x -1)^2 (x+ 2))`


Integrate the rational function:

`(2x - 3)/((x^2 -1)(2x + 3))`


Integrate the rational function:

`(5x)/((x + 1)(x^2 - 4))`


Integrate the rational function:

`(x^3 + x + 1)/(x^2 -1)`


Integrate the rational function:

`2/((1-x)(1+x^2))`


Integrate the rational function:

`1/(x^4 - 1)`


Integrate the rational function:

`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]


Integrate the rational functions `1/(x(x^4 - 1))`


Integrate the rational functions `1/(e^x -1)`[Hint: Put ex = t]


Choose the correct answer `int (dx)/(x(x^2 + 1))` equal

A. `log |x| - 1/2 log (x^2 + 1) + C`

B. `log |x| + 1/2 log(x^2 + 1) + C`

C. `- log|x| + 1/2 log (x^2 + 1) + C`

D. `1/2 log|x| + log(x^2 + 1)+ C`


Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`


Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`


Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`


Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`


Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`


Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`


Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`


Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`


Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`


Integrate the following w.r.t. x : `(1)/(sinx + sin2x)`


Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`


Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`


Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`


Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`


Choose the correct options from the given alternatives :

If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =


Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`


Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`


Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`


Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`


Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`


Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`


Integrate the following w.r.t.x : `x^2/((x - 1)(3x - 1)(3x - 2)`


Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`


Integrate the following w.r.t.x :  `sec^2x sqrt(7 + 2 tan x - tan^2 x)`


Integrate the following w.r.t.x : `(x + 5)/(x^3 + 3x^2 - x - 3)`


Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`


Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx


Evaluate: `int 1/("x"("x"^5 + 1))` dx


Evaluate: `int 1/("x"("x"^"n" + 1))` dx


Choose the correct alternative from the following.

`int "dx"/(("x - 8")("x + 7"))`= 


State whether the following statement is True or False.

If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.


Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx


`int ((2x - 7))/sqrt(4x- 1)  "d"x`


`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`


`int x^7/(1 + x^4)^2  "d"x`


`int x^2sqrt("a"^2 - x^6)  "d"x`


`int 1/(x(x^3 - 1)) "d"x`


`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`


`int 1/(4x^2 - 20x + 17)  "d"x`


`int (sinx)/(sin3x)  "d"x`


`int 1/(2 +  cosx - sinx)  "d"x`


`int sec^2x  "d"x`


`int sin(logx)  "d"x`


`int sec^2x sqrt(tan^2x + tanx - 7)  "d"x`


`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`


`int "e"^x ((1 + x^2))/(1 + x)^2  "d"x`


`int (x^2 + x -1)/(x^2 + x - 6)  "d"x`


`int (3x + 4)/sqrt(2x^2 + 2x + 1)  "d"x`


`int x^3tan^(-1)x  "d"x`


`int (x + sinx)/(1 - cosx)  "d"x`


`int  x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))  "d"x`


`int ("d"x)/(x^3 - 1)`


`int (5"e"^x)/(("e"^x + 1)("e"^(2x) + 9))  "d"x`


`int 1/(sinx(3 + 2cosx))  "d"x`


`int (sin2x)/(3sin^4x - 4sin^2x + 1)  "d"x`


Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?


Choose the correct alternative:

`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =


`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c


State whether the following statement is True or False:

For `int (x - 1)/(x + 1)^3  "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2


Evaluate `int (2"e"^x + 5)/(2"e"^x + 1)  "d"x`


Evaluate `int x^2"e"^(4x)  "d"x`


`int x/((x - 1)^2 (x + 2)) "d"x`


`int 1/(4x^2 - 20x + 17)  "d"x`


If `int(sin2x)/(sin5x  sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______


If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______


Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x2 = t


Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`


Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin x)`


Evaluate the following:

`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`


Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`


Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t2)


The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.


Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`


Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`


If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)


Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.


`int 1/(x^2 + 1)^2 dx` = ______.


If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.


If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1  x/2 + B tan^-1(x/3) + C`, then A – B = ______.


If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.


Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`


Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.


Evaluate`int(5x^2-6x+3)/(2x-3)dx`


Evaluate: 

`int 2/((1 - x)(1 + x^2))dx`


Evaluate.

`int (5x^2 - 6x + 3) / (2x -3) dx`


Share
Notifications



      Forgot password?
Use app×