Integrate the following functions w.r.t. x : x29-x6 - Mathematics and Statistics

Advertisements
Advertisements
Sum

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

Advertisements

Solution

Let I = `int x^2/sqrt(9 - x^6).dx`

Put x3 = t
∴ 3x2 dx = dt

∴ x2dx = `(1)/(3)dt`

∴ I = `int 1/sqrt(9 - t^2).dt/(3)`

= `(1)/(3) int dt/sqrt(3^2 - t^2)`

= `(1)/(3) sin^-1(t/3)  + c`

= `(1)/(3)sin^-1(x^3/3)  + c`.

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.2 (A) [Page 110]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Exercise 3.2 (A) | Q 1.23 | Page 110

RELATED QUESTIONS

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


Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`


Evaluate :`intxlogxdx`


Find : `int((2x-5)e^(2x))/(2x-3)^3dx`


Find : `int(x+3)sqrt(3-4x-x^2dx)`


Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`


Find `intsqrtx/sqrt(a^3-x^3)dx`


Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.


Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`


Integrate the functions:

sin x ⋅ sin (cos x)


Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 


Integrate the functions:

`x/(e^(x^2))`


Integrate the functions:

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


Integrate the functions:

tan2(2x – 3)


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

`1/(cos^2 x(1-tan x)^2`


Integrate the functions:

`sqrt(sin 2x) cos 2x`


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

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


`(10x^9 + 10^x log_e 10)/(x^10 + 10^x)  dx` equals:


`int (dx)/(sin^2 x cos^2 x)` equals:


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`


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

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

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

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

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

Write a value of

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

Write a value of

\[\int \tan^3 x \sec^2 x \text{ dx }\].

 


Write a value of

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

 


Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]


Write a value of\[\int a^x e^x \text{ dx }\]


Write a value of\[\int\frac{\cos x}{\sin x \log \sin x} dx\]

 


Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]


Write a value of

\[\int\frac{a^x}{3 + a^x} \text{ dx}\]

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


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

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

 

 


The value of \[\int\frac{1}{x + x \log x} dx\] is


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

\[\int x \sin^3 x\ dx\]

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


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


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


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


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


Evaluate the following integrals : `int (sin2x)/(cosx)dx`


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


Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 


Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`


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


Evaluate the following integrals:

`int x/(x + 2).dx`


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


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


Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`


Integrate the following functions w.r.t. x : sin4x.cos3x


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


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


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


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 : `(1)/(sinx.cosx + 2cos^2x)`


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 : cos7x


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


Evaluate the following : `int (1)/sqrt(2x^2 - 5).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)/sqrt(x^2 + 8x - 20).dx`


Evaluate the following : `int sinx/(sin 3x).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 + 2sin x - cosx)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 - sinx).dx`


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


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


Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`


Evaluate the following integrals : `int sqrt((9 - x)/x).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 f x^x (1 + log x)*dx`


Choose the correct options from the given alternatives : 

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


Choose the correct options from the given alternatives :

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


Integrate the following w.r.t.x : `(3x + 1)/sqrt(-2x^2 + x + 3)`


Evaluate the following.

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


Evaluate the following.

∫ (x + 1)(x + 2)7 (x + 3)dx


Evaluate the following.

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


Evaluate the following.

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


Evaluate the following.

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


Evaluate the following.

`int 1/(sqrt(3"x"^2 + 8))` dx


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


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


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


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


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


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


`int (cos2x)/(sin^2x)  "d"x`


`int cos^7 x  "d"x`


`int(log(logx))/x  "d"x`


`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1))  "d"x`


To find the value of `int ((1 + logx))/x` dx the proper substitution is ______


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)`


State whether the following statement is True or False:

`int3^(2x + 3)  "d"x = (3^(2x + 3))/2 + "c"`


State whether the following statement is True or False:

`int sqrt(1 + x^2) *x  "d"x = 1/3(1 + x^2)^(3/2) + "c"`


Evaluate `int(3x^2 - 5)^2  "d"x`


If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______ 


`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "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[ tan (log x) + sec^2 (log x)] dx= ` ______


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


`int sec^6 x tan x   "d"x` = ______.


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


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 sqrt(x^2 - a^2)/x dx` = ______.


The value of `sqrt(2) int (sinx  dx)/(sin(x - π/4))` is ______.


`int cos^3x  dx` = ______.


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


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


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


Evaluate the following.

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


Evaluate the following

`int1/(x^2 +4x-5)dx`


Evaluate.

`int(5"x"^2 - 6"x" + 3)/(2"x" - 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 x sqrt(1 + x^2)  dx`


Prove that:

`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.


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

`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).


`int 1/(sin^2x cos^2x)dx` = ______.


Evaluate:

`int(cos 2x)/sinx dx`


Evaluate:

`intsqrt(3 + 4x - 4x^2)  dx`


`int (cos4x)/(sin2x + cos2x)dx` = ______.


Evaluate the following.

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


Evaluate the following:

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


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


Evaluate the following.

`int1/(x^2+4x-5)dx`


Share
Notifications



      Forgot password?
Use app×