Advertisements
Advertisements
Question
Evaluate the following:
`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`
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)`
Evaluate : `intsin(x-a)/sin(x+a)dx`
Find the integrals of the function:
sin3 x cos3 x
Find the integrals of the function:
`(1-cosx)/(1 + cos x)`
Find the integrals of the function:
cos4 2x
Find the integrals of the function:
`(cos 2x+ 2sin^2x)/(cos^2 x)`
Find the integrals of the function:
`1/(sin xcos^3 x)`
Find the integrals of the function:
sin−1 (cos x)
Find the integrals of the function:
`1/(cos(x - a) cos(x - b))`
Find `int (sin^2 x - cos^2x)/(sin x cos x) dx`
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`
Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x \cdot cosec x}dx\] .
Find `int_ (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `
Find `int_ sin ("x" - a)/(sin ("x" + a )) d"x"`
Find: `int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`
Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.
Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`
Evaluate `int tan^8 x sec^4 x"d"x`
Find `int x^2tan^-1x"d"x`
`int "e"^x (cosx - sinx)"d"x` is equal to ______.
Evaluate the following:
`int tan^2x sec^4 x"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)
`int sinx/(3 + 4cos^2x) "d"x` = ______.
