Integrate the following with respect to the respective variable : (x-2)2x - Mathematics and Statistics

Advertisements
Advertisements
Sum

Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`

Advertisements

Solution

Let I = `int (x - 2)^2 sqrt(x)*dx`

= `int (x^2 - 4x + 4)sqrt(x)*dx`

= `int (x^(5/2) - 4x^(3/2) + 4x^(1/2))*dx`

= `int x^(5/2)*dx - 4 int x^(3/2)*dx + 4 int x^(1/2)*dx`

= `x^(7/2)/((7/2)) - 4 x^(5/2)/((5/2)) + 4 x^(3/2)/((3/2))`

= `(2)/(7)x^(7/2) - 8/5x^(5/2) + (8)/(3)x^(3/2) + c`.

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Miscellaneous Exercise 3 [Page 150]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 2.1 | Page 150

RELATED QUESTIONS

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`


Evaluate :`intxlogxdx`


Find : `int((2x-5)e^(2x))/(2x-3)^3dx`


Find : `int(x+3)sqrt(3-4x-x^2dx)`


Find `intsqrtx/sqrt(a^3-x^3)dx`


Integrate the functions:

`(2x)/(1 + x^2)`


Integrate the functions:

`x/(9 - 4x^2)`


Integrate the functions:

`e^(tan^(-1)x)/(1+x^2)`


Integrate the functions:

tan2(2x – 3)


Integrate the functions:

sec2(7 – 4x)


Integrate the functions:

`(2cosx - 3sinx)/(6cos x + 4 sin x)`


Integrate the functions:

`cos x /(sqrt(1+sinx))`


Integrate the functions:

`sin x/(1+ cos x)`


Integrate the functions:

`(sin x)/(1+ cos x)^2`


Integrate the functions:

`1/(1 - tan x)`


Integrate the functions:

`(1+ log x)^2/x`


Integrate the functions:

`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`


Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`


Evaluate: `int (2y^2)/(y^2 + 4)dx`


Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`


Evaluate: `int (sec x)/(1 + cosec x) dx`


\[\int\sqrt{x^2 + x + 1} \text{ dx}\]

\[\int\sqrt{x - x^2} dx\]

\[\int\sqrt{1 + x - 2 x^2} \text{ dx }\]

\[\int e^x \sqrt{e^{2x} + 1} \text{ dx}\]

\[\int\sqrt{9 - x^2}\text{ dx}\]

\[\int\sqrt{16 x^2 + 25} \text{ dx}\]

\[\int\sqrt{4 x^2 - 5}\text{ dx}\]

Write a value of

\[\int\frac{1 + \cot x}{x + \log \sin x} \text{ dx }\]

Write a value of

\[\int e^x \left( \sin x + \cos x \right) \text{ dx}\]

 


Write a value of

\[\int e^{\text{ log  sin x  }}\text{ cos x}. \text{ dx}\]

 Write a valoue of \[\int \sin^3 x \cos x\ dx\]

 


Write a value of\[\int\left( e^{x \log_e \text{  a}} + e^{a \log_e x} \right) dx\] .


Write a value of\[\int e^{ax} \sin\ bx\ dx\]


Write a value of\[\int e^{ax} \cos\ bx\ dx\].

 


Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .


Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]


Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].


`int "dx"/(9"x"^2 + 1)= ______. `


Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`


Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`


Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`


Evaluate the following integrals : tan2x dx


Evaluate the following integrals : `int sin x/cos^2x dx`


Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`


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 : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`


Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`


Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`


Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`


Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`


Integrate the following functions w.r.t. x : x9.sec2(x10)


Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1 


Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`


Integrate the following functions w.r.t.x:

`(2sinx cosx)/(3cos^2x + 4sin^2 x)`


Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`


Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`


Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`


Integrate the following functions w.r.t. x : `cosx/sin(x - a)`


Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`


Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`


Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`


Integrate the following functions w.r.t. x : tan5x


Integrate the following functions w.r.t. x : sin5x.cos8x


Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`


Evaluate the following : `int (1)/(4x^2 - 3).dx`


Evaluate the following : `int (1)/(7 + 2x^2).dx`


Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`


Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`


Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`


Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`


Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`


Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`


Evaluate the following : `int (1)/(4 + 3cos^2x).dx`


Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`


Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`


Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`


Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`


Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`


Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`


Evaluate the following integrals :  `int (3x + 4)/sqrt(2x^2 + 2x + 1).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` =


Choose the correct options from the given alternatives :

`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =


Choose the correct options from the given alternatives : 

`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =


`int logx/(log ex)^2*dx` = ______.


Choose the correct options from the given alternatives :

`int (cos2x - 1)/(cos2x + 1)*dx` =


Choose the correct options from the given alternatives :

`int (e^(2x) + e^-2x)/e^x*dx` =


integrate the following with respect to the respective variable : `x^2/(x + 1)`


Evaluate the following.

`int (1 + "x")/("x" + "e"^"-x")` dx


Evaluate the following.

`int 1/(4"x"^2 - 20"x" + 17)` dx


Evaluate the following.

`int 1/(sqrt("x"^2 -8"x" - 20))` dx


Fill in the Blank.

`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______


Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx


`int 1/sqrt((x - 3)(x - 2))` dx = ________________


`int 1/(cos x - sin x)` dx = _______________


`int x^2/sqrt(1 - x^6)` dx = ________________


`int sqrt(x^2 + 2x + 5)` dx = ______________


`int  ("e"^x(x - 1))/(x^2)  "d"x` = ______ 


`int (sin4x)/(cos 2x) "d"x`


`int logx/x  "d"x`


`int ("e"^(2x) + "e"^(-2x))/("e"^x)  "d"x`


`int 1/(xsin^2(logx))  "d"x`


`int cos^7 x  "d"x`


Choose the correct alternative:

`int(1 - x)^(-2) dx` = ______.


`int (7x + 9)^13  "d"x` ______ + c


State whether the following statement is True or False:

If `int x  "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`


`int (1 + x)/(x + "e"^(-x))  "d"x`


`int(5x + 2)/(3x - 4) dx` = ______


`int (cos x)/(1 - sin x) "dx" =` ______.


If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.


General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)


`int(sin2x)/(5sin^2x+3cos^2x)  dx=` ______.


`int sec^6 x tan x   "d"x` = ______.


`int ("d"x)/(x(x^4 + 1))` = ______.


`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.


`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.


If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.


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 (sin  (5x)/2)/(sin  x/2)dx` is equal to ______. (where C is a constant of integration).


`int(log(logx) + 1/(logx)^2)dx` = ______.


`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.


The value of `intsinx/(sinx - cosx)dx` equals ______.


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 ______.


The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.


`int secx/(secx - tanx)dx` equals ______.


Evaluate `int(1+ x + x^2/(2!)) dx`


Evaluate the following.

`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`


Evaluate.

`int(5"x"^2 - 6"x" + 3)/(2"x" - 3)  "dx"`


Solve the following 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.

`int 1/(x^2 + 4x - 5)dx`


`int x^3 e^(x^2) dx`


Evaluate:

`int 1/(1 + cosα . cosx)dx`


Evaluate `int (1)/(x(x - 1))dx`


Evaluate.

`int (5x^2 - 6x + 3)/(2x - 3) 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.

`int(1)/(x^2 + 4x - 5)dx`


Evaluate the following.

`int x^3 e^(x^2) dx`


Evaluate:

`intsqrt(3 + 4x - 4x^2)  dx`


Evaluate:

`int sin^3x cos^3x  dx`


The value of `int dx/(sqrt(1 - x))` is ______.


Evaluate the following:

`int (1) / (x^2 + 4x - 5) dx`


Evaluate `int (1 + x + x^2/(2!)) dx`


Evaluate `int1/(x(x-1))dx` 


Evaluate `int(1+x+x^2/(2!))dx`


Evaluate `int (1 + "x" + "x"^2/(2!))`dx


Evaluate the following.

`int "x"^3/sqrt(1 + "x"^4)` dx


Share
Notifications



      Forgot password?
Use app×