Advertisements
Advertisements
Question
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Advertisements
Solution
Let I = `int 1/(sqrt(3"x"^2 + 8))` dx
`int 1/(sqrt((sqrt3"x")^2 + (sqrt8)^2))` dx
`= (log |sqrt3"x" + sqrt((sqrt3"x")^2 + (sqrt8)^2)|)/sqrt3` + c
∴ I = `1/sqrt3 log |sqrt3"x" + sqrt(3"x"^2 + 8)|` + c
Alternate method:
Let I = `"I" = int 1/sqrt(3"x"^2 + 8) "dx" = 1/sqrt3 int 1/(sqrt ("x"^2 + 8/3)` dx
`= 1/sqrt3 int 1/sqrt("x"^2 + ((2sqrt2)/sqrt3)^2)` dx
`= 1/sqrt3 log |"x" + sqrt("x"^2 + ((2sqrt2)/sqrt3)^2)| + "c"_1`
`= 1/sqrt3 log |"x" + sqrt("x"^2 + 8/3)| + "c"_1`
`= 1/sqrt3 log |(sqrt3"x" + sqrt(3"x"^2 + 8))/sqrt3| + "c"_1`
`= 1/sqrt3 log|sqrt3"x" + sqrt(3"x"^2 + 8)| - 1/sqrt3 log sqrt3 + "c"_1`
∴ I = `1/sqrt3 log |sqrt3"x" + sqrt(3"x"^2 + 8)|` + c
where c = `"c"_1 - 1/sqrt3 log sqrt3`
Notes
The answer in the textbook is incorrect.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Find `intsqrtx/sqrt(a^3-x^3)dx`
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Evaluate: `int 1/(x(x-1)) 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) :}`
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).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)/(4x + 5x^-11)`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
`int logx/(log ex)^2*dx` = ______.
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)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"` =
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int 1/(cos x - sin x)` dx = _______________
`int logx/x "d"x`
`int 1/(xsin^2(logx)) "d"x`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
`int x^3"e"^(x^2) "d"x`
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int(7x - 2)^2dx = (7x -2)^3/21 + 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 (x + sinx)/(1 + cosx)dx` is equal to ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
`int cos^3x dx` = ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
`int secx/(secx - tanx)dx` equals ______.
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^3/(sqrt(1 + x^4))dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate:
`int sqrt((a - x)/x) dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate `int(1+x+(x^2)/(2!))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(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) 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).
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
What is the value of \[\int\sin^3x\cos^2x\,dx\]?
When applying substitution, what must always be rewritten in terms of the new variable?
For indefinite integrals, what should be done after integration in the new variable?
