Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Write a value of
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following function w.r.t. x:
x9.sec2(x10)
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 : tan5x
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).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)`
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int 1/(xsin^2(logx)) "d"x`
`int cos^7 x "d"x`
`int(1 - x)^(-2) dx` = ______.
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
`int (1 + x)/(x + "e"^(-x)) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int(5x + 2)/(3x - 4) dx` = ______
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int (cos x)/(1 - sin x) "dx" =` ______.
`int 1/(sinx.cos^2x)dx` = ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int cos^3x dx` = ______.
`int (logx)^2/x dx` = ______.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
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`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate `int(1+x+x^2/(2!))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 `int1/(x(x - 1))dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
What is \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
Integration by substitution is the reverse process of which rule?
