Advertisements
Advertisements
प्रश्न
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Advertisements
उत्तर
Let I = `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
We perform actual division and express the result as:
`"Dividend"/"Divisor" = "Quotient" + "Remainder"/"Divisor"`
x - 1
`2"x"^2 - "x" - 10)overline(2"x"^3 - 3"x"^2 - 9"x" + 1)`
`2"x"^3 - "x"^2 - 10"x"`
(-) (+) (+)
`- 2"x"^2 + "x" + 1`
`- 2"x"^2 + "x" + 10`
(+) (-) (-)
- 9
∴ I = `int("x - 1" + (-9)/(2"x"^2 - "x" - 10))` dx
`= int "x" * "dx" - int 1 * "dx" - 9 int 1/(2"x"^2 - "x" - 10) "dx"`
Here 2x2 - x - 10
`= 2("x"^2 + 1/2"x" + 1/16 - 5 - 1/16)`
`= 2 [("x" - 1/4)^2 - 81/16]`
∴ I = `int "x" * "dx" - int 1 * "dx" - 9/2 int 1/(("x" - 1/4)^2 - (9/4)^2)`dx
`= "x"^2/2 - "x" - 9/2 * 1/(2 (9/4)) log |("x" - 1/4 - 9/4)/("x" - 1/4 + 9/4)| + "c"_1`
`= "x"^2/2 - "x" - log |("x" -5/2)/("x + 2")| + "c"_1`
`= "x"^2/2 - "x" - log|("2x" - 5)/(2("x + 2"))| + "c"_1`
`= "x"^2/2 - "x" + log|(2("x + 2"))/("2x" - 5)| + "c"_1`
`= "x"^2/2 - "x" + log |("x + 2")/("2x - 5")| + log 2 + "c"_1`
∴ I = `"x"^2/2 - "x" + log|("x + 2")/("2x - 5")| + "c" "where" "c" = "c"_1 + log 2`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int xcos^3x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
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 ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
