Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Advertisements
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
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Solve:
dy/dx = cos(x + y)
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*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)`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: `int "x" * "e"^"2x"` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int (cos2x)/(sin^2x) "d"x`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int dx/(1 + e^-x)` = ______
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int x^2/sqrt(1 - x^6)dx` = ______.
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int 1/(x(x-1))dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
