Advertisements
Advertisements
Question
Evaluate the following: `int sinx/(sin 3x).dx`
Advertisements
Solution
Let I = `int sinx/(sin 3x).dx`
= `int sinx/(3sinx - 4sin^3x).dx`
= `int (sinx)/(sinx(3 - 4sin^2x)).dx`
= `int (1)/(3 - 4sin^2x).dx`
Dividing both numerator and denominator by cos2x, we get
I = `int (sec^2x)/(3sec^2x - 4tan^2x).dx`
= `int (sec^2x)/(3(1 + tan^2x) - 4tan^2x).dx`
= `int (sec^2x)/(3 - tan^2x).dx`
Put tan x = t
∴ sec2x dx = dt
I = `int dt/(3-t^2)`
I = `int dt/((sqrt(3))^2 - t^2)`
= `int1/((sqrt3)^2 - t^2)dt`
= `(1)/(2sqrt(3)) log |(sqrt(3) + t)/(sqrt(3) - t)| + c`
= `(1)/(2sqrt(3)) log |(sqrt(3) + tanx)/(sqrt(3) - tanx)| + c`.
RELATED QUESTIONS
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Write a value of
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
The value of \[\int\frac{1}{x + x \log x} dx\] is
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int 1/(4"x"^2 - 20"x" + 17)` dx
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate `int 1/((2"x" + 3))` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int (log x)/(log ex)^2` dx = _________
`int (cos2x)/(sin^2x) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
`int sin^-1 x`dx = ?
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
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 ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluated 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 (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int1/(x(x - 1))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
