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_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Evaluate :`intxlogxdx`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`(1+ log x)^2/x`
Write a value of
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate: ∫ |x| dx if x < 0
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int 1/(cos x - sin x)` dx = _______________
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int cos sqrtx` dx = _____________
`int (cos2x)/(sin^2x) "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate the following
`int1/(x^2 +4x-5)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int "cosec"^4x dx` = ______.
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Integration by substitution is the reverse process of which rule?
