Question

How do you solve `int_0^1 sqrt(5x+4) dx`?

Answer 1

`38/15`

Explanation:

We need to use a u-substitution to solve this question.

STEP 1: Identify the u
`u = 5x+4`

STEP 2: Find du
`du = 5 dx`

STEP 3: Change the x-bounds to u-bounds
`x = 0 -> u = 5(0)+4 = 4`
`x = 1 -> u = 5(1)+4 = 9`

STEP 4: Do the u-substitution
`int_0^1 sqrt 5x+4 dx`
`=int_4^9 sqrt(u) * 1/5 du`
remember: we found `du = 5dx`, so if we solve for `dx` we get `dx=1/5 du`
`=1/5 int_4^9 sqrt(u) du`
`=1/5 [2/3 u^(3/2)]_4^9`
`=2/15[u^(3/2)]_4^9`
`=2/15(3^3 - 2^3)`
`=2/15(19)`
`=38/15`