Compute the indefinite integral Use a trigonometric substitution with

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

We compute integrals involving square roots of sums and differences.

The following are the Pythagorean Trigonometric Identities (named for Pythagoras of Samos) which hold for all angles, , in the domains of the functions involved: and

example 1 Compute the indefinite integral The key to the solution is to replace the
variable, , with a trigonometric function in order to eliminate the square root. Since
and the radicand, , have the same algebraic form, we will try the substitution . To
apply this to the integral, we also need to compute the differential, . Substituting
gives, The point of this substitution is that the square root can be computed by
using a Pythagorean trigonometric identity. From there, we can compute the integral
using the skills developed in the section on trigonometric integrals. We have
At this point we have an answer in terms of the variable , but we require the variable
. We will use the triangle below to convert back to .

From the triangle, we see that , and we substitute to get the final answer:

Technically, . However, in the substitution , we can choose the angle, , to be in the
interval . This way, , and .

(problem 1a)

Compute the indefinite integral Use a trigonometric substitution with

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

(problem 1b)

Compute the indefinite integral Use a trigonometric substitution with

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 2 Compute the indefinite integral We choose the substitution so that . The integral becomes

From the triangle above, we can see that so that:

(problem 2)

Compute the indefinite integral Use a trigonometric substitution with

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 3 Compute the indefinite integral We choose the substitution so that . The integral becomes

From the triangle above, we can see that and so that:

(problem 3a)

Compute the indefinite integral Use a trigonometric substitution with

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

(problem 3b)

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 4 Compute the indefinite integral We will use the substitution , so that
.

The integral becomes

From the triangle above, we can see that and so that:

Technically, . However, in the substitution we choose the angle to be in the interval
or . This way, and .

(problem 4)

Compute the indefinite integral Use a trigonometric substitution with

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 5 Compute the indefinite integral We will use the substitution , so that
.

The integral becomes

From the triangle above, we can see that so that:

(problem 5) Compute the indefinite integral Use a trigonometric substitution
with

Substituting these and simplifying the integrand gives

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 6 Compute the indefinite integral We will use the substitution , so that
.

The integral becomes

(problem 6) Compute the indefinite integral Use a trigonometric substitution
with

Substituting these and simplifying the integrand gives

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution , the final answer is

example 7 Compute the indefinite integral We will use the substitution , so that
.

The integral becomes

From the triangle above, we can see that so that:

(problem 7) Compute the indefinite integral

Use a trigonometric substitution with

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 8 Compute the indefinite integral We will use the substitution , so that
.

The integral becomes

From the triangle above, we can see that so that:

(problem 8) Compute the indefinite integral Use a trigonometric substitution
with

Substituting these and simplifying the integrand gives

Substituting these and simplifying the integrand gives

Computing this integral gives

Using the identity, , we can rewrite this as

Based on the substitution ,

The final answer is

example 9 Compute the indefinite integral: We will use the
substitution . Then the differential, . Substituting these gives

From the triangle above, we see that . Back-substituting gives

Technically, . However, in our substitution, , we can specify that the angle be in the
interval . This way, and .

(problem 9) Compute the indefinite integral Use a trigonometric substitution
with

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 10 Compute the indefinite integral We will let . Then the differential . Substituting these, we get

From the triangle above, we can see that and so

(problem 10) Compute the indefinite integral Use a trigonometric substitution
with

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

example 11 Compute the indefinite integral Let . Then . Substituting these, the integral becomes,

From the triangle above, we can see that . Back substitution gives

Note that the absolute value bars were removed because for any . Also note that the was removed from inside the log because and can be combined with the constant of integration.(problem 11) Compute the indefinite integral Use a trigonometric substitution
with

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

The differential is

Substituting these and simplifying the integrand gives

Computing this integral gives Based on the substitution ,

The final answer is

Here is a detailed, lecture style video on trigonometric substitution:

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