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Unformatted text preview: easier to manipulate the integral
using a change of variable.
T HEOREM 1.2 [S UBSTITUTION R ULE ]. If is continuous on and is continuous on the range of ,
2 WATERLOO SOS E XAM -AID: MATH138 M IDTERM
The important thing to note here is that we must change the bounds of integration, and we need to consider
the term . If we are solving an indeﬁnite integral, we must change back to the original variables. If we
are solving a deﬁnite integral, we have two options:
(i) Solve the problem in the new variable, change back to the original variable, and evaluate at the
original bounds; or
(ii) Solve the problem in the new variable, change the bounds of integration, and evaluate at these
new bounds. 2 Techniques for Integration (7.1 - 7.4) We will discuss the all various methods of integration, before looking at examples in the next section.
Typically, you will not be told what method of integration to use on the exam, so you must get used to
Integration by Parts:
Recall the product rule for differentiation: if and are differentiable functions, then
Taking integrals, using the Fundamental Theorem, we get
Rearranging, Another form you may be familiar with is obtained when we use the notation and , to get
Remember the bounds if you are dealing with a deﬁnite integral:
Integration by parts typically works well when you are confronted with a product of functions, but
sometimes can be applied when there is no clear product of functions, by taking . This “1” trick
is useful when the integrand can differentiate to a nice form when multiplied by , for example , since
Another useful trick is to use integration by parts twice to end up with a formula involving our original
integral, then using simple algebra to ﬁnd the solution. This “recurring integral” trick is useful when the
integrand involves a product of functions which are cylic in their derivatives/antderivatives, for example
, since taking the second derivative/integral of is and taking the second derivative/integral of
Trigonometric Integration: 3 WATERLOO SOS E XAM -AID: MATH138 M IDTERM
When you have an integral involving certain combinations of trigonometric functions, it can be useful
to use a substitution such as or to attempt to transform the integrand into a rational
polynomial. We do this by using trigonometric identities.
For combinations of powers of and :
(i) If the power of sine is odd, say we have a term of the form , then we use the identity
Making the substitution , we get to pick up the extra factor of , and
we are left with a polynomial.
(ii) Similarly, if the power of cosine is odd, say we have a term of the form , then we use
the identity to get
Making the substitution , we get ...
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