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Integrate each function by using the table in Appendix $D$.$$\int \sin ^{3} 4 x \cos ^{3} 4 x d x$$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 11

Integration by Use of Tables

Integrals

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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So here on this problem we need to evolve it an indefinite integral which is integration sign cube four X. Because I Ncube four X. B. X. And to evaluate this problem we need to refer the formulas that is given in the appendix city of the book. Now if you ever want formula which is given in the appendix T. And the formula is integration sign cube you Because in cuba you do you it is equals two signed to the power. M plus one. You consigned to the power and minus one. You divide by M plus N. Plus the uh this n minus one divided by empress in integration signed to the power. M. U. Consigned to the power and minus to you deal. So this is a form Lord that we're going to use. We've elevated. Given in trouble. Now if you compare this formula with the given in trouble we can say that you we have four X. And do you would be equal to four dx. Here we have only dx. So to convert into a standard format we can do one thing. We can multiply it with four like this and divide it with four so that it will become four dx. And I've divided it was so that the integration doesn't change. So now we can use exactly this formula to evaluate this ventricle. So and one more thing that we have M equals 23 and n is also three. Now let us put all the values in the formula. So it will be equals two. One by four. This one by four will remain as it is this one no sign to the power M plus and plus one. So um is we have a three. So it would be signed to the power for you. We have four X. So this would be four x times co sign and minus one would be to you. We have four X divide by this would become six so n plus one divide by employees. And so it would be three minus one is +22 divide by six integration signed to the power. M you sign em we have three sign cube you we have four X times consigned to the power and minus two. So three ministries one so you we have four X. So this would be consigned forex. And do you we have four dx next we're going to evaluate this integral first so and to evaluate this integral we can use the same formula this one and in the second case we have M equals 23 and we have one. You we have four X. And do you would be equals two four D X. So I'm going to use the same formula and we'll get this one is a cold too. We are evaluating this in google first. So let me write her. This is integration sign cube four X. Consigned four X times for the X. So we are evaluating the central so uh here and we have one. So if you plug in one hair this will become one management zero. So this whole town would be zero. So we have to consider only this time. So this would be assigned to the power M plus ones and plus one. We have force assigned to the power for you. We have four X divide where unpleasant, unpleasant and we have four. So this is the result. We don't have to consider this time because this n minus one. So one cousin, one minus one is zero. So we can contribute to the consider only this one. So this is the village that we were playing And now you can plug it here. Surat will get so we can put it here and it will be equals to this one by four as it is. And here we have signed to the power forex. This is four X times cosine square four X divide by six plus. This would become 1/3 times this value. We have this much. So this is a sign to the power 44 X divide by four. Now can simplify this and finally we'll get our answer to be won over 48 signed to the power four four x times to assign square four X plus one. Now we have to add the concept of integration. See because the integral is indefinite integral. So that is why we have to add this constancy. So this is a required answer for the given problem. I hope you have understood the problem. Thank you.

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