Advertisements
Advertisements
प्रश्न
`int (dx)/(x(x^2 + 1))` equals:
पर्याय
`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
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`1/(x^2 - 9)`
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 following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(2sinx + 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 : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int sec^3x "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x sin2x cos5x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int xcos^3x "d"x`
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
Which partial form corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\]?
What should be checked before beginning partial-fraction decomposition?
Which factorisation is correct for \[x^{2}-5x+6\]?
For \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\], which equation results after clearing denominators?
What must be included for a repeated linear factor?
