Here is the question: Find this integral by inspection.

Getting the answer they expect is simple enough, consider the derivative of which is and then adjust by a factor of

This gives an answer of

However if you adjust the first integral they give you by factoring out and then apply the same technique with instead the answer turns out to be - . Both differentiate to be the same thing, (unless I'm making an obvious mistake so are they both correct?

Please feel free to correct my thinking or maths if it's flawed