Advertisements
Advertisements
Question
Evaluate the following:
`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`
Advertisements
Solution
Let I = `int (sin^6x + cos^6x)/(sin^2x * cos^2x) "d"x`
= `int ((sin^2x)^3 + (cos^2x)^3)/(sin^2x * cos^2x) "d"x`
= `int ((sin^2x + cos^2x)^3 - 3sin^2x cos^2x(sin^2x + cos^2x))/(sin^2x * cos^2x) "d"x` ......[∵ a3 + b3 = (a + b)3 – 3ab(a + b)]
= `int ((1)^3 - 3sin^2x cos^2x * (1))/(sin^2x cos^2x) "d"x`
= `int (1 - 3sin^2x cos^2x)/(sin^2x cos^2x) "d"x`
= `int (1/(sin^2x cos^2x) - (3sin^2x cos^2x)/(sin^2x cos^2x)) "d"x`
= `int (1/(sin^2x + cos^2x) - 3)"d"x`
= `int ((sin^2x + cos^2x)/(sin^2x cos^2x) - 3) "d"x`
= `int [(1/(cos^2x) + 1/(sin^2x)) - 3]"d"x`
= `int (sec^2x + "cosec"^2x - 3) "d"x`
= `int sec^2x "d"x + int "cosec"^2x "d"x - 3 int 1"d"x`
= tan x – cot x – 3x + C
Hence, I = tan x – cot x – 3x + C.
APPEARS IN
RELATED QUESTIONS
Evaluate :`int_(pi/6)^(pi/3) dx/(1+sqrtcotx)`
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:
sin3 x cos3 x
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 x - sinx)/(1+sin 2x)`
Find the integrals of the function:
tan4x
Find the integrals of the function:
`(sin^3 x + cos^3 x)/(sin^2x cos^2 x)`
Find the integrals of the function:
`1/(sin xcos^3 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))`
`int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx` is equal to ______.
`int (e^x(1 +x))/cos^2(e^x x) dx` equals ______.
Find `int (sin^2 x - cos^2x)/(sin x cos x) dx`
Find `int dx/(x^2 + 4x + 8)`
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_ sin ("x" - a)/(sin ("x" + a )) d"x"`
Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.
Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`
Find `int "dx"/(2sin^2x + 5cos^2x)`
`int "e"^x (cosx - sinx)"d"x` is equal to ______.
`int (sin^6x)/(cos^8x) "d"x` = ______.
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
`int sinx/(3 + 4cos^2x) "d"x` = ______.
`int (cos^2x)/(sin x + cos x)^2 dx` is equal to
