I'm not certain what "by inspection" means but I suspect it means there is an obvious substitution. You would be able to do it easily if it were only , wouldn't you? Okay, make that happen!
Evaluate the following integral by inspection
How would one solve this by inspection? The only thing I can think of is some sort of application of the product rule. But that doesn't sound like a process you'd do by inspection. :/
You should recall that: and if you actually have which you can rewrite and use in that integral in the form of .
You should always bear in mind that you could (and should be able to) use differential calculus just as you would use algebraic manipulation in order to simplify the integral.
What I demonstrated is a 'technique' sometimes called 'introducing terms under the differentiation sign' and substitution is a method of performing that 'technique' when terms are too complicated to 'introduce them under the differentiation sign' without abbreviating the notation by introducing a new variable. Only when you actually introduce a new variable (by denoting the substitution) you can speak of the method of substitution.
The biggest difference is when calculating definite integrals. When using the substitution method one has to ajust the limits of the integration according to the substitution used, while when performing the 'introducing terms under the differentiation sign' technique you are always in terms of the original variable (most often called x) and thus the limits of the integration remain the same.