Advertisements
Advertisements
प्रश्न
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Advertisements
उत्तर
Let I = `int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Put sin x = t
∴ cosx dx = dt
∴ I = `int 3/(4t^2 + 4t - 1)dt`
I = `3/4 int 1/(t^2 + t - 1/4)dt`
I = `3/4 int 1/((t^2 + t + 1/4) - 1/4 - 1/4)dt`
I = `3/4 int 1/ ((t + 1/2)^2 - 1/2)dt`
I = `3/4 int 1/sqrt((t + 1/2)^2 - (1/sqrt2)^2)dt`
`[∵ int 1/(x^2 - a^2)dx = 1/(2a) log |(x - a)/(x + a)| + c]`
I = `3/4 xx 1/(2(1/sqrt2)) log |(t + 1/2 - 1/sqrt2)/(t + 1/2 + 1/sqrt2)| + c`
I = `3/(4sqrt2) log |(2sqrt2t + (2sqrt2)/2 - (2sqrt2)/sqrt2)/(2sqrt2t + (2sqrt2)/2 - (2sqrt2)/sqrt2)| + c`
I = `3/(4sqrt2) log |(2sqrt2t + sqrt2 - 2)/(2sqrt2t +sqrt2 + 2)| + c`
I = `3/(4sqrt2) log |(2sqrt2sin + sqrt2 - 2)/(2sqrt2sin +sqrt2 + 2)| + c`
APPEARS IN
संबंधित प्रश्न
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
sec2(7 – 4x)
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Evaluate the following integrals:
tan2x dx
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int (log x)/(log ex)^2` dx = _________
`int sqrt(1 + sin2x) dx`
`int(1 - x)^(-2) dx` = ______.
`int (7x + 9)^13 "d"x` ______ + c
`int sin^-1 x`dx = ?
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate `int (1)/(x(x - 1))dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
