Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Advertisements
उत्तर
Let I = `int(5 - 3x)(2 - 3x)^(-1/2).dx`
Put 2 – 3x = t
∴ –3dx = dt
∴ dx = `(-dt)/(3)`
Also, x = `(2 - t)/(3)`
∴ I = `int[5 - 3((2 - t)/3)]t^(-1/2).((-dt)/(3))`
= `-1/3(5 - 2 + t)t^(-1/2)dt`
= `-1/3 int(3 + t)t^(-1/2) dt`
= `-1/3 int(3t^(-1/2) + t^(1/2))dt`
= `-3/3 int t^(-1/2)dt - (1)/(3) int t^(1/2) dt`
= `-t^(1/2)/((1/2)) - (1)/(3).t^(3/2)/((3/2)) + c`
= `-2sqrt(2 - 3x) - (2)/(9)(2 - 3x)^(3/2) + c`
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int \log_e x\ dx\].
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
`int sqrt(1 + "x"^2) "dx"` =
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
`int cot^2x "d"x`
`int sin^-1 x`dx = ?
`int1/(4 + 3cos^2x)dx` = ______
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
Integration by substitution is the reverse process of which rule?
