Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`
Advertisements
उत्तर
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
संबंधित प्रश्न
Evaluate : `intsin(x-a)/sin(x+a)dx`
Find the integrals of the function:
cos 2x cos 4x cos 6x
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:
cos4 2x
Find the integrals of the function:
`(sin^2 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:
tan3 2x sec 2x
Find the integrals of the function:
tan4x
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:
`(cos 2x)/(cos x + sin x)^2`
Find the integrals of the function:
`1/(cos(x - a) cos(x - b))`
Find `int dx/(x^2 + 4x + 8)`
Differentiate : \[\tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right)\] with respect to x .
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_ (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_ (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.`
`int "dx"/(sin^2x cos^2x)` is equal to ______.
`int (sin^6x)/(cos^8x) "d"x` = ______.
Evaluate the following:
`int ((1 + cosx))/(x + sinx) "d"x`
Evaluate the following:
`int tan^2x sec^4 x"d"x`
Evaluate the following:
`int sqrt(1 + sinx)"d"x`
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.
`int sinx/(3 + 4cos^2x) "d"x` = ______.
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
