Advertisements
Advertisements
Question
Find `int dx/sqrt(sin^3x cos(x - α))`.
Advertisements
Solution
Let I = `int dx/sqrt(sin^3x cos(x - α))`
= `int dx/sqrt((sin^4x)/sinx [cosx cosα + sinx sinα]`
= `int dx/(sin^2 xsqrt(cotx cosα + sinα)`
= `int ("cosec"^2x dx)/sqrt(cotx cosα + sinα)`
Let cot x cos α + sin α = t
Then, dt = – cosec2 x cos α dx
∴ I = `int (-dt)/(cosαsqrt(t))`
= `(-2sqrt(t))/cosα + C`
= `(-2sqrt(cotx cosα + sinα))/cosα + C`
= `- 2 sec αsqrt(cotx cosα + sinα) + C`.
RELATED QUESTIONS
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Write a value of\[\int a^x e^x \text{ dx }\]
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
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 : sin5x.cos8x
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate `int 1/(x (x - 1))` dx
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int 1/(sin^2x cos^2x)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-1))dx`
