Advertisements
Advertisements
Question
Find `int_ (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `
Advertisements
Solution
According to question,
let I = `int_ (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `
I = `int_ (sin "x" - cos "x")/sqrt(sin^2 "x" + cos^2 "x" + 2 sin "x" .cos "x") d"x"`
= `int_ ( sin "x" - cos "x")/sqrt((sin"x" + cos "x")^2 d"x"`
= `int_ (sin "x" - cos "x")/(sin "x" + cos "x") d"x"`
let sin x + cos x = t
⇒ (cos x - sin x) dx = dt
I = `int_ (-1)/("t") d"t"`
= -ln t + C
= ln `(1/"t") + "C"`
⇒ I = ln `((1)/(sin "x" + cos "x")) + "C"`
APPEARS IN
RELATED QUESTIONS
Evaluate :`int_(pi/6)^(pi/3) dx/(1+sqrtcotx)`
Evaluate : `intsin(x-a)/sin(x+a)dx`
Find the integrals of the function:
sin2 (2x + 5)
Find the integrals of the function:
sin 3x cos 4x
Find the integrals of the function:
sin3 (2x + 1)
Find the integrals of the function:
sin 4x sin 8x
Find the integrals of the function:
`(1-cosx)/(1 + cos x)`
Find the integrals of the function:
sin4 x
Find the integrals of the function:
`(cos 2x - cos 2 alpha)/(cos x - cos alpha)`
Find the integrals of the function:
`(cos 2x+ 2sin^2x)/(cos^2 x)`
Find the integrals of the function:
`(cos 2x)/(cos x + sin x)^2`
Find the integrals of the function:
`1/(cos(x - a) cos(x - b))`
Evaluate `int_0^(3/2) |x sin pix|dx`
Differentiate : \[\tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right)\] with respect to x .
Find `int_ (sin2"x")/((sin^2 "x"+1)(sin^2"x"+3))d"x"`
Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`
Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`
Find: `int sin^-1 (2x) dx.`
Evaluate the following:
`int ((1 + cosx))/(x + sinx) "d"x`
Evaluate the following:
`int ("d"x)/(1 + cos x)`
Evaluate the following:
`int tan^2x sec^4 x"d"x`
Evaluate the following:
`int (sinx + cosx)/sqrt(1 + sin 2x) "d"x`
Evaluate the following:
`int sqrt(1 + sinx)"d"x`
Evaluate the following:
`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.
