Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ sin(2θ) dθ / (1 + cos(2θ))²

Accepted Answer

Welcome back everyone. Evaluate the integral sine to theta d theta divided by 1 minus cosine 2 theta squared. For this problem, before we begin integrating, let's try and simplify the expression that we want to integrate. We can use the double angle identities. So in the numerator, we have sin to theta. In the denominator, we have 1 minus cosine to theta. And then squared starting with the numerator, let's remember that sin 2 theta is 2 sine theta cosine theta. And for cosine to theta. Let's remember that 1 minus cosine to theta. is equal to 2 sin squared theta. So what do we get? Well, essentially we have to square this value which will give us 2 squared. That's 4 sin squared theta squared is sin to the power of 4 of theta. And from here we can simplify 2 divided by 4 is going to give us 1/2. Multiplied by sine divided by sin to the power of 4 leaves us with sine cubed theta in the denominator. And in the numerator we have cosine theta, right? So now this is going to be equal to 1/2 multiplied by. What we can do is rewrite it as cosine theta divided by sine theta and then multiplied by 1 divided by sine squared theta. In other words, we're splitting sine cubed into sine multiplied by sine squared. And this is equal to 1/2 multiplied by cosine divided by sine is cotangent, so we get cotangent theta. One divided by sine is cosequent theta. And since we're dividing by sin squared, it is going to be cosequin squared theta. So we can now rewrite the integral as 1/2. We can factor out the constant. Integral of cotangent. Theta cosi and squared theta. Followed by The Sara. Now, in order to integrate it, we're going to use a substitution. Let's suppose that U is equal to co sequent theta. And let's remember that our next step is to find the derivative of you with respect to theta. The derivative of cosequent theta is negative cosequent theta. Co tangent Sara. And now what we're going to do from here is multiply both sides by negative d theta to show that negative DU is equal to cosequent theta, cotangentheta d theta. Well done, so now we can rewrite the integral in terms of you and the you. First of all, we have our constant one half. Integral Now U is equal to cosequent theta and negative duu is cosequant theta cotangent theta d theta. Notice that we can split cosequin squared into cosequent theta. Multiplied by cosi and theta, right? So it basically tells us that co-sequent theta. One of them is going to be equal to. You And cosequan theta cotangent theta d theta, this part in particular. Is going to become equal. 2 negative view. We can essentially factor out the negative signs. We will have 1/2 integral of you. The u which is now an elementary integral. Its value is -1/2 and to integrate UDU we're going to apply the power rule that's EU 2 divided by 2. Plus a constant of integration c. We get -14 u2 + c, which is equal to -14 multiplied by Since U is cosequon theta, we get -14 co sequin 2 theta plus c. That's our final answer and thank you for watching.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.∫ sin(2θ) dθ / (1 + cos(2θ))²

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sin(2θ) dθ / (1 + cos(2θ))²