1-3 Evaluate the integral using the indicated trigonometric substitution. Sketch and label the a

1-3 Evaluate the integral using the indicated trigonometric...

Key Concepts

Pythagorean Identities

Pythagorean identities, such as sin²? + cos²? = 1, are fundamental in simplifying integrals transformed via trigonometric substitution. These identities allow the expression of radical terms in simpler trigonometric forms, which is a critical step in effectively evaluating the integrals.

Right Triangle Representation

A right triangle is used to visualize the relationship between the original variable and the trigonometric substitution. By constructing a triangle with sides representing the values from the substitution, one can easily express all trigonometric functions and algebraic terms in terms of the original variable, facilitating the process of back substitution after integration.

Differential Computation in Substitution

When a substitution is made, it is necessary to compute the differential of the substituted variable. This step involves differentiating the substitution equation with respect to the new variable, which then replaces the dx in the integral, ensuring that the integral is correctly transformed.

Trigonometric Substitution

Trigonometric substitution is a method used to simplify the integration of expressions involving square roots of quadratic functions. By substituting a variable with a trigonometric function, such as setting x equal to a constant times sine or cosine, the radical expression is transformed using Pythagorean identities, which makes the integral easier to evaluate.

Transcript

00:02 Okay, hello folks.

00:05 Today we'll be evaluating the following trigonometric integral using the following substitution.

00:12 Now, this question does say to draw an associated right triangle, i don't like doing that because that puts an implicit assumption on that the values of theta are between 0 and pi on 2, which is not necessarily true, since x here can vary between negative 3 and positive 3, what i want here, or what i'm going to do here, is i'm going to say x equals 3 sine theta.

00:40 However, we're going to put a restriction on theta to be between pi on 2 to minus pi on 2.

00:52 Okay, and that way we can get all values of sine theta, so going from negative 1 to positive 1, and that means x can take all its positive possible values.

01:04 Okay, so keep this restriction in mind because we're going to need it potentially.

01:09 So let's make the substitution.

01:11 So this will be 27 sine cubed theta, and then this will be square root of 9 minus 9 sine squared theta.

01:24 And then for dx, we have dx, dx, d theta, is 3 cosine of theta.

01:31 So that means dx is 3 cosine of theta d theta.

01:36 So that'll be 3 cosine de theta d theta.

01:41 And that's going to be...

01:43 So i'm going to write just 3 cubed here.

01:46 So this will be 3 to the power of 4, sine cubed theta.

01:51 And...

01:52 Oh, sorry about that.

01:54 We can take another 3 from out the square root, and this will be 1 minus sine squared theta.

02:00 And then there's a cosine theta and then a d theta.

02:07 Okay, good.

02:08 So then this is going to be 3 to the power of 5 sine cubed theta and we have the square root of, so 1 minus sine squared theta is cosine squared theta and then cosine theta d theta.

02:29 So this is going to be 3 to the power of 5 sine cubed theta.

02:34 Now square root of cosine squared theta is the absolute value of cosine theta and another cosine theta d theta.

02:54 So now we have this is 3 to the power of 5.

03:02 So this is going to be sine cubed theta, cosine theta.

03:10 Now what do we do with this cosine theta in the absolute values? this is where we're going to need our restriction on theta.

03:17 So the theta is between minus pi on two and pi on two.

03:20 And for theta taking these values, we know that cosine theta is greater than or equal to zero.

03:28 So that means the absolute value of cosine theta is just cosine theta itself...

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