Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Advertisements
उत्तर
Let I = `int (1)/(x(x^3 - 1)).dx`
= `int (x^-4)/(x^-4x(x^3 - 1)).dx`
= `int (x^-4)/(1 - x^-3).dx`
= `(1)/(3) int (3x^-4)/(1 - x^-3).dx`
= `(1)/(3) int (d/dx(1 - x^-3))/(1 - x^-3).dx`
= `(1)/(3)log|1 - x^-3 | + c ...[∵ int (f'(x))/f(x)dx = log|f(x)| + c]`
= `(1)/(3)log|1 - 1/x^3| + c`
= `(1)/(3)log|(x^3 - 1)/x^3| + c`.
Alternative Method :
Let I = `int (1)/(x(x^3 - 1)).dx`
= `int x^2/(x^3(x^3 - 1)).dx`
Put x3 = t
∴ 3x2dx = dt
∴ x2dx = `dt/(3)`
∴ I = `int (1)/(t(t - 1)).dt/(3)`
= `(1)/(3)int(1)/(t(t - 1))dt`
= `(1)/(3) int(t - (t - 1))/(t(t - 1))dt`
= `(1)/(3) int(1/(t - 1) - 1/t)dt`
= `(1)/(3)[int (1)/(t - 1)dt - int (1)/tdt]`
= `(1)/(3)[log |t - 1| - log|t|] + c`
= `(1)/(3)log|(t - 1)/t| + c`
= `(1)/(3)log|(x^3 - 1)/x^3| + c`.
APPEARS IN
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`1/(1 + cot x)`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
`int "dx"/(9"x"^2 + 1)= ______. `
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
Evaluate: ∫ |x| dx if x < 0
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int sqrt(1 + sin2x) dx`
`int (sin4x)/(cos 2x) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int dx/(1 + e^-x)` = ______
`int(5x + 2)/(3x - 4) dx` = ______
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following
`int1/(x^2 +4x-5)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 the following.
`int1/(x^2+4x-5) dx`
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
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:
`intsqrt(sec x/2 - 1)dx`
