Advertisements
Advertisements
प्रश्न
Solve the following:
`int_0^1 e^(x^2)*x^3dx`
उत्तर
Let I = `int_0^1 e^(x^2)*x^3dx`
= `int_0^1 e^(x^2)*x^2*xdx`
Put x2 = t
∴ 2x·dx = dt
∴ x·dx = `(1)/(2)*"dt"`
When x = 0, t = 0
When x = 1, t = 1
∴ I = `(1)/(2) int_0^1 e^"t"*"tdt"`
= `(1)/(2){["t" int e^"t"*"dt"]_0^1 - int_0^1[d/"dt" ("t") int e^"t"*"dt"]"dt"}`
= `(1)/(2) [["t"*e^"t"]_0^1 - int_0^1 1*e^"t" "dt"]`
= `(1)/(2){(1*e^1 - 0) - [e^"t"]_0^1}`
= `(1)/(2)[e - (e^1 - e^0)]`
= `(1)/(2)(e - e + 1)`
∴ I = `(1)/(2)`.
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int_(-pi/4)^(pi/4) (1)/(1 - sinx)*dx`
Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`
Evaluate the following : `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following definite integral:
`int_4^9 (1)/sqrt(x)*dx`
Evaluate the following definite integrals: If `int_0^"a" (2x + 1)*dx` = 2, find the real value of a.
Evaluate the following definite integrals: `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))*dx`
Evaluate the following integrals : `int_0^1 log(1/x - 1)*dx`
Solve the following : `int_4^9 (1)/sqrt(x)*dx`
Solve the following : `int_0^1 (1)/(2x - 3)*dx`
Choose the correct alternative:
`int_0^"a" 3x^5 "d"x` = 8, then a =
State whether the following statement is True or False:
`int_0^"a" 3x^2 "d"x` = 27, then a = 2.5
`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = ______.
`int_(-5)^5 log ((7 - x)/(7 + x))`dx = ?
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
Evaluate the following definite integral:
`int_-2^3 1/(x+5) *dx`
Solve the following:
`int_1^3 x^2 log x dx`
Evaluate the following definite integrals: `int_1^2 (3x)/((9x^2 - 1))*dx`
Solve the following.
`int_0^1e^(x^2) x^3 dx`
