Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Advertisements
उत्तर
Let I = `int (1)/(3 - 2cos 2x).dx`
Put tan x = t
∴ x = tan–1 t
∴ dx = `dt/(1 + t^2) and cos2x = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(3 - 2((1 - t^2)/(1 + t^2))).dt/(1 + t^2)`
= `int (1 + t^2)/(3 + 3t^2 - 2 + 2t^2).dt/(1 + t^2)`
= `int (1)/(1 + 5t^2)dt`
= `(1)/(5) int (1)/((1 /sqrt(5))^2 + t^2)dt`
= `(1)/(5) xx (1)/((1/sqrt(5)))tan^-1(t/(1/sqrt(5))) + c`
= `(1)/sqrt(5)tan^-1(sqrt(5)tanx) + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Integrate the functions:
`1/(1 - tan x)`
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
`int "dx"/(9"x"^2 + 1)= ______. `
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
If f'(x) = `x + 1/x`, then f(x) is ______.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate `int(1 + x + x^2/(2!))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
`int (cos4x)/(sin2x + cos2x)dx` = ______.
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int (1 + x + x^2/(2!)) 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`
What is \[\int\cot t\,dt\] in the evaluation of \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
When applying substitution, what must always be rewritten in terms of the new variable?
