Advertisements
Advertisements
प्रश्न
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Advertisements
उत्तर
`int (1)/sqrt(3x^2 + 8).dx`
= `(1)/sqrt(3) int (1)/sqrt(x^2 + 8/3).dx`
= `(1)/sqrt(3) int (1)/sqrt(x^2 + (sqrt(8/3))^2).dx`
= `(1)/sqrt(3) log |x + sqrt(x^2 + (sqrt(8/3))^2)| + c_1`
= `(1)/sqrt(3) log |x + sqrt(x^2 + 8/3)| + c_1`
= `(1)/sqrt(3) log |(sqrt(3)x + sqrt(3x^2 + 8))/sqrt(3)| + c_1`
= `(1)/sqrt(3) log |sqrt(3)x + sqrt(3x^2 + 8)| - logsqrt(3) + c_1`
= `(1)/sqrt(3) log |sqrt(3)x + sqrt(3x^2 + 8)| + c, "where" c = c_1 - logsqrt(3)`
Alternative Method :
`int (1)/sqrt(3x^2 + 8).dx`
= `int (1)/sqrt((sqrt(3)x)^2 + (sqrt(8))^2).dx`
= `(log|sqrt(3)x + sqrt((sqrt(3)x)^2 + sqrt((8))^2| + c))/sqrt(3)`
= `(1)/sqrt(3) log |sqrt(3)x + sqrt(3x^2 + 8)| + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
`int "dx"/(9"x"^2 + 1)= ______. `
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals:
tan2x dx
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: `int "x" * "e"^"2x"` dx
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int x/(x + 2) "d"x`
`int dx/(1 + e^-x)` = ______
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int 1/(sinx.cos^2x)dx` = ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int1/(x(x-1))dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Which standard substitution is used for \[\sqrt{\mathrm{a}^2-x^2}\], \[\frac{1}{\sqrt{\mathrm{a}^2-x^2}}\], or \[\mathrm{a}^2-x^2\]?
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
What is \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
