Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Advertisements
उत्तर
Let I = `int sec^2x sqrt(7 + 2 tan x - tan^2x) *dx`
Put tan x = t
∴ sec2x·dx = dt
∴ I = `int sqrt(7 + 2t - t^2)*dt`
= `int sqrt(7 - (t^2 - 2t))*dt`
= `int sqrt(8 - (t^2 - 2t + 1))*dt`
= `int sqrt((2sqrt(2))^2 - (t - 1)^2)*dt`
= `((t - 1)/2) sqrt((2sqrt(2))^2 - (t - 1)^2) + ((2sqrt(2))^2)/(2) sin^-1((t - 1)/(2sqrt(2))) + c`
= `((t - 1)/2) sqrt(7 + 2t - t^2) + 4sin^-1 ((t - 1)/(2sqrt(2))) + c`
= `((tanx - 1)/2)sqrt(7 + 2tanx - tan^2x) + 4sin^-1 ((tanx - 1)/(2sqrt(2))) + c`.
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
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 : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
`int "dx"/(("x" - 8)("x" + 7))`=
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int (2x - 7)/sqrt(4x- 1) dx`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "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^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int (x + sinx)/(1 - cosx) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int xcos^3x "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int (dx)/(2 + cos x - sin 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)
`int 1/(x^2 + 1)^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?
Integration by partial fractions is a method used to integrate which functions?
Which partial form corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\]?
Which partial-fraction decomposition is used for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})^{2}}\]?
Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?
What is done after long division, before writing the appropriate partial-fraction decomposition?
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 numerator should be used for each distinct linear factor in a partial-fraction decomposition?
