Advertisements
Advertisements
प्रश्न
Find the integrals of the function:
tan4x
Advertisements
उत्तर
Let `I = int tan^4 x dx = int (sec^2 x - 1)^2 dx`
`= (sec^4 x - 2 sec^2 x + 1) dx`
`= int sec^4 x dx - 2 int sec^2 x dx + int 1 dx`
`= int sec^4 x dx - 2 tan x + x + C_1`
⇒ `I = I_1 - 2 tan x + x + C_1` ...(i)
Where `I_1 = intsec^4 x dx`
Now, `I_1 = int sec^4 x dx = int sec^2 x * sec^2 x dx`
`= int (1 + tan^2 x) sec^2 x dx.`
Put tan x = t
⇒ sec2 x dx = dt
∴ `I_1 = int (1 + t^2) dt = t = t^3/3 + C_2`
`= tan x + 1/3 tan^3 x + C_2` .....(ii)
From (i) and (iii), we have,
`I = tan x + 1/3 tan^3 x + C_2 - 2 tan x + x + C_1`
`= 1/3 tan^3 x - tan x + x + C`
APPEARS IN
संबंधित प्रश्न
Evaluate :`int_(pi/6)^(pi/3) dx/(1+sqrtcotx)`
Evaluate : `intsin(x-a)/sin(x+a)dx`
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:
`(1-cosx)/(1 + cos x)`
Find the integrals of the function:
`cos x/(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:
tan3 2x sec 2x
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))`
`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 dx/(x^2 + 4x + 8)`
Evaluate: `int_0^π (x sin x)/(1 + cos^2x) dx`.
Differentiate : \[\tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right)\] with respect to x .
Find `int_ (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `
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)`
Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`
Find: `int sin^-1 (2x) dx.`
Evaluate `int tan^8 x sec^4 x"d"x`
`int "e"^x (cosx - sinx)"d"x` is equal to ______.
`int (sin^6x)/(cos^8x) "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 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
