Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
उत्तर
Let I = `int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Put x2 + 6x + 3 = t
∴ (2x + 6) dx = dt
∴ I = `int "dt"/sqrt"t"`
`= int "t"^((-1)/2)`dt
`= "t"^(1/2)/(1/2)` + c
`= 2 sqrt"t"` + c
∴ I = `2 sqrt("x"^2 + "6x" + 3)` + c
Alternate Method:
Let I = `int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
`"d"/"dx" ("x"^2 + "6x" + 3)` = 2x + 6
∴ I = `int ("d"/"dx" ("x"^2 + "6x" + 3))/(sqrt("x"^2 + 6"x" + 3))` dx
∴ I = `2 sqrt("x"^2 + "6x" + 3)` + c ....`[because int ("f" '("x"))/sqrt("f"("x")) "dx" = 2sqrt("f"("x")) + "c"]`
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`1/(1 + cot x)`
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals : `int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
`int x^3"e"^(x^2) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
