Advertisements
Advertisements
Question
Integrate the following functions w.r.t.x:
cos8xcotx
Advertisements
Solution
Let I = `int cos^8xcotxdx`
= `int cos^8x. cosx/sinx .dx`
Put sinx = t
∴ cosx dx = dt
cos8x = (cos2x)4
= (1 – sin2x)4
= (1 – t2)4
= 1 – 4t2 + 6t4 – 4t6 + t8
I = `int(1 - 4t^2 + 6t^4 - 4t^6 + t^8)/tdt`
= `int[1/t - 4t +6t^3 - 4t^5 + t^7]dt`
= `int 1/t dx - 4 int tdt + 6 int t^3 dt - 4 int t^5 dt + int t^7 dt`
= `log|t| - 4 (t^2/2) + 6(t^4/4) - 4(t^6/6) + t^8/(8) + c`
= `log|sinx| - 2sin^2x + 3/2 sin^4x - 2/3 sin^6x + (sin^8x)/(8) + c`.
RELATED QUESTIONS
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : tan5x
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int dx/(1 + e^-x)` = ______
`int (cos x)/(1 - sin x) "dx" =` ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
`int secx/(secx - tanx)dx` equals ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
After choosing \[u=g(x)\], what is the next step?
Integration by substitution is the reverse process of which rule?
