Advertisements
Advertisements
प्रश्न
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
उत्तर
Let I = `int (1)/(cos2x + 3sin^2x).dx`
= `int (1)/(1 - 2sin^2x + 3sin^2x).dx`
= `int(1)/(1 + sin^2x).dx`
Dividing both numerator and denominator by cos2x, we get
I = `int(sec^2x dx)/(sec^2x + tan^2x)`
= `int (sec^2x dx)/(1 + tan^2x + tan^2x)`
= `int (sec^2x dx)/(2tan^2x + 1)`
Put tan x = t
∴ sec2x dx = dt
∴ I = `int (1)/(2t^2 + 1)dt`
= `(1)/(2) int (1)/(t^2 + (1/sqrt(2))^2)dt`
= `(1)/(2) xx (1)/((1/sqrt(2)))tan^-1 (t/(1/sqrt(2))) + c`
= `(1)/sqrt(2)tan^-1 (sqrt(2)tan x) + c`.
APPEARS IN
संबंधित प्रश्न
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
`int "dx"/(9"x"^2 + 1)= ______. `
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals : `int sin 4x cos 3x dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : `(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(4"x"^2 - 20"x" + 17)` dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int cos sqrtx` dx = _____________
`int (log x)/(log ex)^2` dx = _________
`int (sin4x)/(cos 2x) "d"x`
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int x^x (1 + logx) "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int dx/(1 + e^-x)` = ______
`int1/(4 + 3cos^2x)dx` = ______
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate:
`int 1/(1 + cosα . cosx)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
