smc-2564-20-Polynomial

Using a suitable substitution, `int_a^b (x^3e^(2x^4))\ dx`, where `a` and `b` are real constants, can be written as

A. `int_a^b (e^(2u))\ du`

B. `int_(a^4)^(b^4) (e^(2u))\ du`

C. `1/8 int_a^b (e^u)\ du`

D. `1/4 int_(a^4)^(b^4) (e^(2u))\ du`

E. `1/8 int_(8a^3)^(8b^3) (e^u)\ du`

Show Answers Only

`D`

Show Worked Solution

`u`	`= x^4`
`(du)/(dx)`	`= 4x^3\ \ =>\ \ 1/4\ du = x^3\ dx`

`text(When)\ \ x=a, \ u=a^4`

`text(When)\ \ x=b, \ u=b^4`

`:. int_a^b x^3 e^(2x^4)\ dx`

`= 1/4 int_(a^4)^(b^4) e^(2u)\ du`

`=> D`