Question

Using the U- Substitution u=sqrt(2x), integral form 2-8 dx/ sqrt(2x) + 1 is equivalent to ...

I'm not quite sure how to solve using u substitution.

Help please!

Answer 1

We will use u-substitute:
`u= √(2x) , (du)/(dx)= (1)/( √(2x) )= (1)/(u)`Then for substitution:dx=u du. and integral becomes:
`\int { (u)/(u+1) } \, du = \int { (u+1-1)/(u+1) } \, du= \int{1} \, du- \int { (1)/(u-1) } \, dx`=u-ln(u+1)=
`√(2x)-ln( √(2x)+1)`. Now we will change the values of limits:
`√(16)-ln( √(16)+1)-( √(4)-ln( √(4)+1))`=4-ln(5)-2+ln(3)=2+ln(0,6)=2-0.51=1.49