Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int_0^1 x^3"e"^(-2x) "d"x`
उत्तर
Bernoulli’s formula,
`int "uv" "d"x` = uv1 – u1v2 + u2v3 – u3v4 + ….
u = x3, v = e–2x
u1 = 3x2, v1 = `("e"^(-2x))/(-2)`
u2 = 6, v2 = `("e"^(-2x))/4`
u3 = 6, v3 = `("e"^(-2x))/(-8)`
u4 = 0, v3 = `("e"^(- 2x))/16`
`int_0^1 x^3 "e"^(-2x) "d"x = [(x^3"e"^(-2))/(-2) - (3x^2"e"^(-2x))/4 + (6x "e"^(-2x))/(-8) - (6"e"^(-2x))/16]_0^1`
`int_0^1 x^3 "e"^(-2x) "d"x = "e"^(-2x)[x^3/(- 2) - (3x^2)/4 + (6x)/(-8) -6/16]_0^1`
= `"e"^(-2)[- 1/2 - 6/4 - 3/8] - "e"^0 (- 3/8)`
= `"e"^(-2) ((-4 - 12 - 3)/8) + 3/8`
= `3/8 - 19/8 "e"^(-2)`
∴ `int_0^1 x^3"e"^(-2x) "d"x = 3/8 - 19/(8"e"^2)`
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
`int_0^1 (sin(3tan^-1 x)tan^-1 x)/(1 + x^2) "d"x`
Evaluate the following:
`int_0^(1/sqrt(2)) ("e"^(sin^-1x) sin^-1 x)/sqrt(1 - x^2) "d"x`
Evaluate the following:
`int_0^(pi/2) x^2 cos 2x "d"x`
Choose the correct alternative:
If `f(x) = int_0^x "t" cos "t" "dt"`, then `("d"f)/("d"x)` =
