Evaluate the following. ∫ (x + 1)(x + 2)7 (x + 3)dx - Mathematics and Statistics

Advertisements
Advertisements
Sum

Evaluate the following.

∫ (x + 1)(x + 2)7 (x + 3)dx

Advertisements

Solution

Let I = ∫ (x + 1)(x + 2)7 (x + 3)dx

Put x + 2 = t

∴ dx = dt

Also, x = t - 2

∴ x + 1 = t - 2 + 1

= t - 1

and x + 3 = t - 2 + 3

= t + 1

∴ I = `int ("t" - 1) * "t"^7 ("t" + 1) * "dt"`

`= int ("t"^2 - 1) * "t"^7 * "dt"`

`= int ("t"^9 - "t"^7) "dt"`

`= int "t"^9 "dt" - int "t"^7 "dt"`

`= "t"^10/10 - "t"^8/8 + "c"`

∴ I = `("x + 2")^10/10 - ("x + 2")^8/8` + c

  Is there an error in this question or solution?
Chapter 5: Integration - EXERCISE 5.2 [Page 123]

APPEARS IN

RELATED QUESTIONS

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`


 
 

Evaluate :

`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`

 
 

Integrate the functions:

`1/(x + x log x)`


Integrate the functions:

`1/(x-sqrtx)`


Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 


Integrate the functions:

`e^(2x+3)`


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions `sin x/(1+ cos x)`


Integrate the functions in `(sin x)/(1+ cos x)^2`


Integrate the functions in `((x+1)(x + logx)^2)/x`


Choose the correct answer `int (dx)/(sin^2 x cos^2 x)` equals

(A) tan x + cot x + C

(B) tan x – cot x + C

(C) tan x cot x + C

(D) tan x – cot 2x + C


Evaluate : `∫1/(3+2sinx+cosx)dx`


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`


\[\int\sqrt{3 + 2x - x^2} \text{ dx}\]

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

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

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

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

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

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

Write a value of

\[\int x^2 \sin x^3 \text{ dx }\]

Write a value of

\[\int\frac{\cos x}{3 + 2 \sin x}\text{  dx}\]

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

 


Write a value of\[\int \cos^4 x \text{ sin x dx }\]


Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]


Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]


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


Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]


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


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


\[\text{ If } \int\left( \frac{x - 1}{x^2} \right) e^x dx = f\left( x \right) e^x + C, \text{ then  write  the value of  f}\left( x \right) .\]

\[\int\frac{\cos^5 x}{\sin x} \text{ dx }\]

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


 Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`


Integrate the following w.r.t. x : x3 + x2 – x + 1


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


Integrate the following w.r.t. x : `2x^3 - 5x + 3/x + 4/x^5`


Evaluate the following integrals : tan2x dx


Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 


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


Evaluate the following integrals : `int tanx/(sec x + tan x)dx`


Evaluate the following integrals : `int sin 4x cos 3x dx`


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


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


Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`


Evaluate the following integrals:

`int (sin4x)/(cos2x).dx`


Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`


Evaluate the following integrals : `int cos^2x.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 : `(1 + x)/(x.sin (x + log x)`


Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`


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


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


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 : sin4x.cos3x


Integrate the following functions w.r.t. x : `(10x^9  10^x.log10)/(10^x + x^10)`


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


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


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

`(5 - 3x)(2 - 3x)^(-1/2)`


Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`


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 : `(1)/(2 + 3tanx)`


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

cos8xcotx


Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`


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


Evaluate the following : `int (1)/(25 - 9x^2).dx`


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


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


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


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((9 - x)/x).dx`


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


Evaluate the following : `int (logx)2.dx`


Choose the correct option from the given alternatives : 

`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =


Choose the correct options from the given alternatives :

`int (e^x(x - 1))/x^2*dx` =


Choose the correct options from the given alternatives :

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


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


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


Evaluate the following.

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


Evaluate the following.

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


Evaluate the following.

`int x/(4x^4 - 20x^2 - 3)dx`


Choose the correct alternative from the following.

`int "dx"/(("x" - "x"^2))`= 


`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c


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: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).


Evaluate: `int 1/(sqrt("x") + "x")` dx


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


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


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


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


If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______


`int sqrt(1 + sin2x)  "d"x`


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


`int logx/x  "d"x`


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


`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`


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


`int x^x (1 + logx)  "d"x`


`int 1/(xsin^2(logx))  "d"x`


`int sqrt(x)  sec(x)^(3/2) tan(x)^(3/2)"d"x`


`int cot^2x  "d"x`


`int cos^7 x  "d"x`


`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)`


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


`int sin^-1 x`dx = ?


If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.


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


`int1/(4 + 3cos^2x)dx` = ______ 


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


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


`int sec^6 x tan x   "d"x` = ______.


`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.


`int (f^'(x))/(f(x))dx` = ______ + c.


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 dx/(2 + cos x)` = ______.

(where C is a constant of integration)


Write `int cotx  dx`.


`int (logx)^2/x dx` = ______.


Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.


Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.


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


Evaluate the following.

`int 1/(x^2+4x-5)  dx`


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)


Evaluate:

`int sqrt((a - x)/x) dx`


Evaluate:

`int(sqrt(tanx) + sqrt(cotx))dx`


If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)


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:

`intsqrt(3 + 4x - 4x^2)  dx`


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


Evaluate the following.

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


Evaluate the following.

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


Evaluate `int1/(x(x-1))dx` 


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


Evaluate the following.

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


Evaluate the following.

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


Share
Notifications



      Forgot password?
Use app×