Advertisements
Advertisements
Question
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Advertisements
Solution
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
RELATED QUESTIONS
Find: `I=intdx/(sinx+sin2x)`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`1/(x^4 - 1)`
`int (xdx)/((x - 1)(x - 2))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
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 : `(3x - 2)/((x + 1)^2(x + 3)`
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 : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int sqrt(4^x(4^x + 4)) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (sinx)/(sin3x) "d"x`
`int sin(logx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int xcos^3x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int x^2"e"^(4x) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
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 (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + 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 (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
