Advertisements
Advertisements
Question
Find: `int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`
Advertisements
Solution
`int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`
Put 2 + sin x = t
⇒ 1 + sin x = t - 1
cos x dx = dt
`int_ ("dt")/(("t" -1) "t")`
= `int_ ((1)/("t" - 1) - (1)/("t")) "dt"`
= `int_ (1)/("t" -1) "dt" - int 1/"t" "dt"`
= log (t - 1) - log t + C
= log (2 + sin x - 1) - log (2 + sin x) + C
= log (1 + sin x) - log (2 + sin x) + C
= `"log" ((1+ sin "x")/(2 + sin "x")) + "C" ` ...`(∵ "log m" - "log n" = "log" ("m"/"n"))`
APPEARS IN
RELATED QUESTIONS
Evaluate : `intsin(x-a)/sin(x+a)dx`
Find the integrals of the function:
sin 3x cos 4x
Find the integrals of the function:
cos 2x cos 4x cos 6x
Find the integrals of the function:
sin x sin 2x sin 3x
Find the integrals of the function:
sin 4x sin 8x
Find the integrals of the function:
`cos x/(1 + cos 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`
`int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx` is equal to ______.
Evaluate `int_0^(3/2) |x sin pix|dx`
Find `int((3 sin x - 2) cos x)/(13 - cos^2 x- 7 sin x) dx`
Find `int_ (log "x")^2 d"x"`
Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration.
Find:
`int"dx"/sqrt(5-4"x" - 2"x"^2)`
Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.
Find `int "dx"/(2sin^2x + 5cos^2x)`
Find `int x^2tan^-1x"d"x`
`int "dx"/(sin^2x cos^2x)` is equal to ______.
Evaluate the following:
`int sqrt(1 + sinx)"d"x`
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
Evaluate the following:
`int sin^-1 sqrt(x/("a" + x)) "d"x` (Hint: Put x = a tan2θ)
The value of the integral `int_(1/3)^1 (x - x^3)^(1/3)/x^4 dx` is
`int (cos^2x)/(sin x + cos x)^2 dx` is equal to
