Question

How do you find the integral `int_0^13dx/(root3((1+2x)^2)` ?

Answer 1

By substitution,
`int_0^{13}1/{root3{(1+2x)^2}}dx=3`

Let `u=1+2x`.
By taking the derivative,
`{du}/{dx}=2`
By taking the reciprocal,
`{dx}/{du}=1/2`
By multiply by `du`,
`dx={du}/2`

Since `x` goes from 0 to 13, `u` goes from 1 to 27.
Now, we can rewrite the integral in terms of `u`.
`int_0^{13}1/{root3{(1+2x)}}dx =int_1^{27}1/{root3{u^2}}{du}/2`
by simplifying,
`=1/2int_1^{27}u^{-2/3}du`
by Power Rule,
`=1/2[3u^{1/3}]_1^{27} =3/2[root3{27}-root3{1}]=3/2cdot 2=3`