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# Find indefinite integrals of powers of sin

From the previous two exercises (here and here) we know

and

Use these results to establish the following formulas:

1. .
2. .
3. .
4. ( Note: There is an error in my edition of the book for part (c). It requests we prove , omitting the in the first term of the result. This is just an error in the book since the formula as stated is false.)

1. Using the second formula, and the triple angle identity for cosine ( which implies , derived in this exercise), we have

2. Using the second formula above and then the first we have (and also recalling the trig identity which implies ),

3. Using the second formula above and then part (a) we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

The function

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

The function

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

The function

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

First, let’s rewrite ,

Then,

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

First, let’s rewrite ,

Then,

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

First, let’s expand ,

Then,

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

First, let’s rewrite ,

Then,

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

First, let’s expand ,

Then,

is a primitive of since

Then, by the second fundamental theorem of calculus we have

# Find a primitive and apply the second fundamental theorem of calculus

Let

Find a function such that (i.e., a primitive of ). Use the second fundamental theorem of calculus to evaluate

The function

is a primitive of since

Then, by the second fundamental theorem of calculus we have