For any prove that
Proof. We make the substitution
Then, evaluating the integral
For any prove that
Proof. We make the substitution
Then, evaluating the integral
So we then have
Here, we change the the name of the variable of integration from to . (We can always rename the variable of integration since integrating is the same as integrating , for example.) So this means we have
Then, using the formula we established in part (a) we have
Now, we use the substitution method, letting
So we obtain,
The last equality follows since so that this is an even function. Hence (by a previous exercise) the integral from -1 to 1 is twice the integral from 0 to
For prove
Proof. First it will help if we simplify the integral:
Then we use the method of substitution, letting
So we have,
For positive integers prove
Proof. Let
Then,
Prove that
Let
These equations imply,
Therefore, substituting for we have,
The final step follows since we can rename the variable of integration (since since all we’ve done is rename the variable)
Define
with positive integers and a real number. Show that
Proof. Let
Then,
Use the method of substitution to prove invariance under translation (Theorem 1.18 on page 81 of Apostol) and to prove expansion or contraction of the interval of integration (Theorem 1.19 on page 81 of Apostol).
Theorem: (Invariance Under Translation) For a function f integrable on an interval [a,b] and for every we have
Proof. If is a primitive of , then
Let
So,
Hence, we indeed have
Theorem: (Expansion or Contraction of the Interval of Integration) For a function f integrable on an interval [a,b] and for every with ,
Proof. Let
Then we have
Thus, we indeed have
Use the method of substitution to evaluate the following integral:
Let’s simplify the integral some,
Now, let
So, we can evaluate,
Use the method of substitution to evaluate the following integral:
First, let’s simplify the integral some
Now, let
Then, we can evaluate the integral
Use the method of substitution to evaluate the following integral:
Let
Then, we can evaluate the integral,