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
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`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int sin x/cos^2x dx`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
`int x^3"e"^(x^2) "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int 1/(sinx.cos^2x)dx` = ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate `int 1/("x"("x" - 1)) "dx"`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int(1+x+x^2/(2!))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 x^3/(sqrt(1 + x^4)) dx`
