Advertisements
Advertisements
प्रश्न
`int x^3tan^(-1)x "d"x`
Advertisements
उत्तर
Let I = `int x^3*tan^(-1)x*"d"x`
= `int (tan^-1x)x^3 "d"x`
= `(tan^-1x) int x^3 "d"x - int["d"/("d"x) (tan^-1x) int x^3 "d"x] "d"x`
= `(tan^-1x)*(x^4/4) - int 1/(1 + x^2)* x^4/4 "d"x`
= `x^4/4 tan^-1x + 1/4 int ((-x^4))/(1 + x^2) * "d"x`
= `x^4/4 tan^-1x + 1/4 int ((1 - x^4) - 1)/(1 + x) "d"x`
= `x^4/4 tan^-1x + 1/4 int ((1 - x^2)(1 + x^2) - 1)/(1 + x^2) "d"x`
= `x^4/4 tan^-1x + 1/4 int (1 - x^2 - 1/(1 + x^2)) "d"x`
= `x^4/4 tan^-1x + 1/4 (x - x^3/3 - tan^-1x) + "c"`
∴ I = `1/4 tan^-1x (x^4 - 1) - x/12 (x^2 - 3) + "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:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sin(logx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int x sin2x cos5x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
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 dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
