The formalway of solvingsuchintegralsis to use -substitutionby letting equalthe denominator. Here,
let and Substituting,
Remark:The integralmustuse the absolutevaluesymbolbecausealthough may havenegativevalues,
the domainof is restrictedto
Example7:
Evaluate
Solution:
As you can see here,the derivativeof the denominatoris Our numeratoris However,
whenwe multiplythe numeratorby we get the derivativeof the denominator. Hence
Again,we couldhaveused -substitution.
Example8:
Evaluate.
Solution:
To solve,we rewritethe integrandas
Lookingat the denominator, its derivativeis. So we needto inserta minussign in the numerator: