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 (sinx)/sqrt(36-cos^2x)dx`
Evaluate :`intxlogxdx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals:
`int (cos2x)/sin^2x 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 : `intsqrt(1 + sin 5x).dx`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate `int 1/("x" ("x" - 1))` dx
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int logx/x "d"x`
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int (cos2x)/(sin^2x) "d"x`
`int(log(logx))/x "d"x`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int sin^-1 x`dx = ?
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int(5x + 2)/(3x - 4) dx` = ______
`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) = ______.
`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(log(logx) + 1/(logx)^2)dx` = ______.
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` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
`int secx/(secx - tanx)dx` equals ______.
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate `int1/(x(x - 1))dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
`int x^3 e^(x^2) 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`
Evaluate the following.
`intx sqrt(1 +x^2) 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 the following.
`int1/(x^2+4x-5)dx`
