Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
Advertisements
उत्तर
Let I = `int sqrt(tanx) "d"x`
Put tan x = t2
⇒ sec2x dx = 2t dt
∴ I = `int "t" * (2"t")/(sec^2x) "dt"`
= `2 int "t"^2/(1 + "t"^4) "dt"`
= `int (("t"^2 + 1) + ("t"^2 - 1))/((1 + "t"^4)) "dt"`
= `int ("t"^2 + 1)/(1 + "t"^4) "dt" + int ("t"^2 - 1)/(1 + "t"^4) "dt"`
= `int (1 + 1/"t"^2)/("t"^2 + 1/"t"^2) "dt" + int (1 - 1/"t"^2)/("t"^2 + 1/"t"^2) "dt"`
= `int (1 + 1/"t"^2)/(("t" - 1/"t")^2 + 2)"dt" + int (1 - 1/"t"^2)/(("t" + 1/"t")^2 - 2)"dt"`
Put u = `"t" - 1/"t"`
⇒ du = `(1 + 1/"t"^2)"dt"` in first integral
And put v = `"t" + 1/"t"`
⇒ dv = `(1 - 1/"t"^2)"dt"` in second integral
∴ I = `int "du"/("u"^2 + (sqrt(2)^2)) + int "dv"/("v"^2 - (sqrt(2)^2))`
= `1/sqrt(2) tan^-1 "u"/sqrt(2) + 1/(2sqrt(2)) log|("v" - sqrt(2))/("v" + sqrt(2))| + "C"`
= `1/sqrt(2) tan^-1 ("t" - 1/"t")/sqrt(2) + 1/(2sqrt(2)) log |("t" + 1/"t" - sqrt(2))/("t" + 1/"t" + sqrt(2))| + "C"`
= `1/sqrt(2) tan^-1 ("t"^2 - 1)/(sqrt(2)"t") + 1/(2sqrt(2)) log |("t"^2 + 1 - sqrt(2)"t")/("t"^2 + 1 + sqrt(2)"t")| + "C"`
= `1/sqrt(2) tan^-1 ((tanx - 1)/sqrt(2tan x)) + 1/(2sqrt(2)) log |(tan x - sqrt(2 tanx) + 1)/(tan x + sqrt(2 tan x) + 1)| + "C"`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
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 : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(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"d"x)/(x^4 - x^2 - 12)`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
