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 : `int x^2/((x^2+2)(2x^2+1))dx`
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)(x - 3))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
`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.
`int sqrt(4^x(4^x + 4)) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int x^3tan^(-1)x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate `int x log x "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`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))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
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
