Advertisements
Advertisements
प्रश्न
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
Advertisements
उत्तर
Let `I = int(5x^2 + 4x +7)/(2x + 3)^(3/2) dx`
Put 2x + 3 = t2 ...(i)
Differentiating w.r.t. x, we get
2dx = 2t dt
∴ dx = t dt
From (i), we get
`x = (t^2 - 3)/2`
∴ `I = int (5((t^2 - 3)/2)^2 + 4((t^2 - 3)/2) + 7)/(t^2)^(3/2) * t dt`
`I = int (5((t^4 - 6t^2 + 9)/4) + 2t^2 - 6 + 7)/t^3 * t dt`
`I = int (5t^4 - 30t^2 + 45 + 8t^2 + 4)/(4t^3) * t dt`
`I = int (5t^4 - 22t^2 + 49)/(4t^2) dt`
`I = 5/4 int t^2 dt - 22/4 int dt + 49/4 int t^(-2) dt`
`I = 5/4 * t^3/3 - 22/4 t + 49/4 * t^(-1)/(-1) + c`
`I = 5/12t^3 - 11/2t - 49/(4t) + c`
∴ `I = 5/12(2x + 3)^(3/2) - 11/2 sqrt(2x + 3) - 49/4 * 1/sqrt(2x + 3) + c`
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find :
`∫ sin(x-a)/sin(x+a)dx`
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 : `(3x - 2)/((x + 1)^2(x + 3)`
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: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
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 : `(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)`
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x sin2x cos5x "d"x`
`int ("d"x)/(x^3 - 1)`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x log x "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
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.
`int 1/(x^2 + 1)^2 dx` = ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + 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 = ______.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
