Advertisements
Advertisements
Question
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Advertisements
Solution
Let I = `int (1)/(4 + 3cos^2x).dx`
Dividing both numerator and denominator by cos2x, we get
I = `int (sec^2x)/(4sec^2 x + 3).dx`
= `int (sec^2x)/(4(1 + tan^2x) + 3).dx`
= `int (sec^2x)/(4tan^2x + 7).dx`
Put tan x = t
∴ sec2x dx = dt
I = `int dt/(4t^2 + 7)`
= `int dt/((2t)^2 + (sqrt(7))^2`
= `(1)/sqrt(7)tan^-1 ((2t)/sqrt(7)).(1)/(2) + c`
= `(1)/(2sqrt(7))tan^-1 ((2tanx)/sqrt(7)) + c`.
APPEARS IN
RELATED QUESTIONS
Evaluate :`intxlogxdx`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Write a value of
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
`int "dx"/(9"x"^2 + 1)= ______. `
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
`int logx/(log ex)^2*dx` = ______.
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int cos^7 x "d"x`
`int sin^-1 x`dx = ?
`int1/(4 + 3cos^2x)dx` = ______
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
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 ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
