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# Prove an integral identity for even integer powers of cosine

For any prove that

Proof. We make the substitution

Then, evaluating the integral

# Prove another identity of the integral of some trig functions

1. Prove the following identity:

2. Using part (a) deduce the formula

1. Proof. Following the hint, we make the substitution

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

2. Now to deduce the requested formula, first we use the trig identity , which implies , to rewrite the integral

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

# Prove an identity for integrals of trig functions

For prove

Proof. First it will help if we simplify the integral:

Then we use the method of substitution, letting

So we have,

# Prove an identity of integral equations

For positive integers prove

Proof. Let

Then,

# Prove the validity of a given integral transform

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)

# Evaluate a given integral equation

Define

with positive integers and a real number. Show that

Proof. Let

Then,

# Prove expansion/contraction and translation invariance of interval of integration using method of substitution

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

# Evaluate the integral using substitution

Use the method of substitution to evaluate the following integral:

Let’s simplify the integral some,

Now, let

So, we can evaluate,

# Evaluate the integral using substitution

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

# Evaluate the integral using substitution

Use the method of substitution to evaluate the following integral:

Let

Then, we can evaluate the integral,