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

Trigonometric Substitution

This technique involves replacing the original variable with a trigonometric function to simplify the integrand—especially useful when dealing with square roots of quadratic expressions. In problems like these, setting x = a·tan? transforms algebraic expressions into trigonometric identities, reducing the integral to a form that is often easier to integrate.

Right Triangle Representation

After a trigonometric substitution, a right triangle is used to represent the relationship between the variables. By considering the definition of the trigonometric function (for example, tan? = opposite/adjacent), one can determine the lengths of the sides of the triangle. This geometric interpretation helps derive expressions for other trigonometric functions (like sec?) in terms of x, thereby simplifying the radical expressions that appear in the integrand.

Differential Substitution

When a substitution is made for the variable x, its differential dx must also be expressed in terms of the new variable (here, ?). This step is essential to correctly transform the entire integral from the x-domain to the ?-domain. Computing the derivative of the substitution function (for instance, differentiating x = a·tan? to obtain dx = a·sec^2? d?) ensures that all parts of the integral are consistently converted.

Pythagorean Identities in Integration

Key trigonometric identities, such as 1 + tan^2? = sec^2?, play a critical role after the substitution. These identities allow the radical expressions to be simplified, converting square roots into expressions involving sec? or other trigonometric functions. This simplification is often the turning point that makes the integral tractable.

Transcript

00:04 Okay, folks, so today we're going to evaluate the following integral using the following substitution.

00:13 So, let's get stuck into it.

00:18 So, once again, if you watched one of my other videos, you'll know that i'm not a fan of using the right triangle because it makes implicit assumptions on theta being between zero and pi on two, which is not always the case.

00:34 In fact, for this case, since x here can be taken for all real numbers, here you want theta to be between pi on two, sorry, we don't want to include pi on two, because tan pi on two is not defined, and minus pi on two.

00:55 So, now keep that in mind because we'll need it.

01:01 And taking theta between minus pi on two and pi on two means x can take all real values since tan takes all real values when theta ranges from minus pi on two to pi on two.

01:14 So let's make that substitution.

01:17 So this is going to be three cubed tan cubed theta.

01:23 And then on the bottom we have the square root of nine plus nine tan squared theta.

01:32 And what's d x so d x or dx d theta is going to be three sex squared theta and that means d x is just going to be three sex squared theta d theta okay so here we go this is going to be multiplied by three sex squared theta d theta okay good so then let's continue this is going to be three cubed tan cube theta over the square root.

02:07 So let's bring out that 3.

02:10 And this is going to be the square root of 1 plus tan squared theta.

02:15 And this is 3 .6 squared theta.

02:18 D theta.

02:20 Okay.

02:21 So these 3s cancel out.

02:23 That's nice.

02:25 And then we're left with the integral of 3 cubed tan cubed theta.

02:30 And then this is going to be over the square root of sex squared theta.

02:36 Because 1 plus tan squared theta is sex squared theta.

02:38 And then another sex squared theta, d theta.

02:43 Okay, so then this is going to be the integral of three cubed, tan cubed theta, over the absolute value of seccatheta, sex squared theta, d theta.

02:58 So this is going to be, so now we need to use our restriction.

03:04 So remember, second theta is just one on cosine of theta.

03:08 And when theta is between minus pi on two and pi on two, cosine of theta is always positive, which means sec a theta is always positive.

03:17 Okay, so just to be clear, so sec theta is equal to 1 on cosine of theta, and cosine of theta is positive if theta is between pi on 2 and minus pi on 2.

03:43 Okay, good.

03:44 So that means we can get rid of this absolute value sign.

03:55 Since we know secer theta is always going to be positive for the values of theta we've taken, this is going to be three cubed, tan cubed theta, over sec theta, sec squared theta, d theta.

04:11 Okay, so we can cancel one of those, and now we're left with three cubed, tan cube theta, second theta d theta okay so how can we deal with this so this is a bit tricky but um we can get out of this so let's change everything to sine theta and cosine theta so this is going to be three cubed so this is going to be sine cubed theta over cosine to the power four theta this is because second theta is 1 on cosine a theta.

05:05 Okay.

05:06 And then d theta.

05:09 All right, good.

05:10 So now we can rearrange this a bit.

05:12 This is 3 cubed sine squared theta over cosine 4 to the feeder, sine a theta, d theta.

05:22 So this is going to be 3 cubed sine squared theta.

05:27 So 1 minus cosine squared theta.

05:32 And this is going to be cosine 4 to the feeder...