How do you find the integral ∫e4edxx⋅√ln(x)dx ?
1 Answer
Sep 6, 2014
Since dx has been placed twice, I'm going to assume that the equation should read
To integrate something by substitution (also known as the change-of-variable rule), we need to select a function
To find the integral of your function, we do the following:
- Let
u=ln(x) , thendudx=1x - this is a standard derivative. - Substitute these two new functions into the equation:
∫e4edxx⋅√ln(x)=∫e4e1x⋅udx=∫e4e1u⋅dudxdx - Find new terminals - this is a crucial step, because we're changing the variable!
When x=e,u=ln(e)=1 and when x=e4,u=ln(e4)=4 - Substitute these new terminals in, and "cancel out" the two
dx terms:
∫e4e1u⋅dudxdx=∫41duu=∫41u−1du - Integrate normally. The answer you get for this "new" integral will be exactly the same answer as the original integral.