Question

If q(v) = integrate sqrt(4 + w ^ 5) dw from 0 to pi ^ 1 * tren; q^ prime (v)=

Answer 1

Given:

`q(v)=\int_0^(v^7)√(4+w^5)dw`

Required:

We need to find q'(v).

Step-by-step explanation:

Consider the given equation.

`q(v)=\int_0^(v^7)√(4+w^5)dw`
`q(v)=\int_0^(v^7)√(4+w^5)*(√(4+w^5))/(√(4+w^5))dw`
`q(v)=\int_0^(v^7)(4+w^5)/(√(4+w^5))dw`
`We\text{ know that d\lparen}√(4+w^5))=(1)/(2)(1)/(√(4+w^5))5w^4dw.`
`(2)/(5w^4)\text{ d\lparen}√(4+w^5))=(1)/(√(4+w^5))dw.`

The given integral can be written as follows.

`q(v)=\int_0^(v^7)(4+w^5)(2)/(5w^4)d(√(4+w^5))`
`q(v)=(2)/(5)\int_0^(v^7)((4+w^5))/(w^4)d(√(4+w^5))`
`q(v)=(2)/(5)\int_0^(v^7)((√(4+w^5))^2)/(w^4)d(√(4+w^5))`