Evaluate the integrals in Exercises 37–46.

∫(...

Question

Evaluate the integrals in Exercises 37–46.

∫(sin 2θ - cos 2θ)/(sin 2θ + cos 2θ)³dθ

Accepted Answer

Welcome back everyone. Compute the integral, integral of sin for c minus cosine for c.dc divided by sin for c plus cosine for cq. For this problem we're going to use use substitution. Let's suppose that u is equal to sin of 4 c plus cosine of 4 c. Now our next step is to find the derivative of u with respect to c. So the derivative of sin of 4 t is going to be cosine of 4 t multiplied by the derivative of 4 t according to the chain rule, which is 4. So we get 4 cosine of 4 t. Plus the derivative of cosine of 4 t is going to be negative sine of 4 t first of all multiplied by the derivative of 4 t which is 4. And I'll notice that if we factor out 4, we got 4. Multiplied by Cosine of 4 cminus sin of c and moreover we can factor out the negative sign to get -4 in parenthes positive sign of c minus cosine of 4 c. Now notice if we divide both sides by -4 and multiply both sides by dc, we're going to get 1/4 DU. Equals Sine of word c minus cosine of word cdc. Which is what we have in the numerator of the integral, right? So now we are ready to rewrite our integral as integral of -1/4. DU, this is what we have in the numerator. In the denominator we have you raised to the power of 3, or basically you cubed. Let's go ahead and factor out 1/4, specifically a negative 1/4. This is a constant, and integrate DU divided by u cubed. What we can do is simply apply the power rule and to do that we are going to rewrite this expression as u to the power of -3DU. So now we're going to get -14. Multiplied by u to the power of -3 + 1, which gives us. You to the power of -2 divided by -2. Plessy constant of integration c. We can cancel out the negative signs. We can combine our constants to get 1/8. You raised to the power of -2 can be rewritten as 1 divided by u 2. Plus c, which is equal to 1 divided by 8 u 2+ c and finally we're going to replace u. With the actual expression in terms of tea. So we get 1 divided by 8 multiplied by sin of 4 c plus cosine of 4 c which is raised to the power of 2, it is squared. Plus C. And that will be our final answer for this problem. Let's go ahead and. La Bellette And Perform the final steps just to ensure that we can see that -2 was canceled out before. Specifically, we canceled out the negative sign only. And now we can label our final answer. Thank you for watching.

Evaluate the integrals in Exercises 37–46.∫(sin 2θ - cos 2θ)/(sin 2θ + cos 2θ)³dθ

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Evaluate the integrals in Exercises 37–46.

∫(sin 2θ - cos 2θ)/(sin 2θ + cos 2θ)³dθ