Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Advertisements
उत्तर
Let I = `int (2x)/(4 - 3x - x^2).dx`
Let `(2x)/(4 - 3x - x^2)`
= `(2x)/((4 + x)(1 - x)`
= `"A"/(4 + x) + "B"/(1 - x)`
∴ 2x = A(1 – x) + B(4 + x)
Put 4 + x = 0, i.e. x = – 4, we get
– 8 = A(5) + B(0)
∴ A = `-(8)/(5)`
Put 1 – x = 0, i.e x = 1, we
2 = A(0) + B(5)
∴ B = `(2)/(5)`
∴ `(2x)/(4 - 3x - x^2) = ((-8/5))/(4 + x) + ((2/5))/(1 - x)`
∴ I = `int [((-8)/5)/(4 + x) + ((2/5))/(1 - x)].dx`
= `-(8)/(5) int(1)/(4 + x).dx + (2)/(5) int (1)/(1 - x).dx`
= `-(8)/(5)log|4 + x| + (2)/(5).(log|1 - x|)/(-1) + c`
= `-(8)/(5)log|4 + x| - (2)/(5)log|1 - x| + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(x(x^4 - 1))`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
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 with respect to the respective variable : `(6x + 5)^(3/2)`
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 : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
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 ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (sinx)/(sin3x) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int xcos^3x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
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 (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
