Advertisements
Advertisements
प्रश्न
`int (dx)/(x(x^2 + 1))` equals:
विकल्प
`log |x| - 1/2 log |x^2 + 1| + C`
`log |x| + 1/2 log |x^2 + 1| + C`
`- log |x| + 1/2 log |x^2 + 1| + C`
`1/2 log |x| + log (x^2 + 1) + C`
Advertisements
उत्तर
`log |x| - 1/2 log |x^2 + 1| + C`
Explanation:
Let `I = int dx/(x (x^2 + 1))`
`= int x/(x (x^2 + 1)) dx`
Put x2 = t
2x dx = dt
`I = 1/2 int (2x dx)/(x (x^2 + 1))`
`= 1/2 int dt/(t (t + 1))`
Now, `1/(t (t + 1)) = A/t + B/(t + 1)`
1 = A(t + 1) + Bt
Putting t = 0, 1 = A
∴ A = 1
Putting t = -1, 1 = B(-1)
∴ B = -1
`therefore 1/(t (t + 1)) = 1/t - 1/(t + 1)`
`therefore 1/2 int 1/(t (t + 1)) dt = 1/2 int 1/t dt - 1/2 int 1/(t + 1) dt`
`= 1/2 log abs t - 1/2 log abs (t + 1) + C`
`= 1/2 log abs (x ^2) - 1/2 log abs(x ^2 + 1) + C`
`= log abs x - 1/2 log abs(x^2 + 1) + 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^2+1)(x - 1))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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 : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` 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 x^2sqrt("a"^2 - x^6) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"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^4 - x^2 - 12)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
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 x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int 2/((1 - x)(1 + x^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
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
Which partial-fraction decomposition is used for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})^{2}}\]?
What is done after long division, before writing the appropriate partial-fraction decomposition?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
What are the values of \[\mathrm{A}\] and \[\mathrm{B}\] in \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\]?
What numerator should be used for each distinct linear factor in a partial-fraction decomposition?
What must be included for a repeated linear factor?
