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Integrate each function by using the table in Appendix $D$.$$\int_{0}^{\pi / 3} 3 \sin ^{3} x d x$$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 11

Integration by Use of Tables

Integrals

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Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:44

Use a table of integrals t…

01:57

Evaluate each integral.

01:35

Evaluate each definite int…

02:59

Evaluate the integrals.

02:28

Evaluate the integrals…

02:18

01:24

04:01

04:15

01:23

So here on this problem we have to evaluate and definite integral which is integration zero to pi over three. three. Sign Cube DX. Not to evaluate this interval. We have to refer the appendix city that is given in the book. Now let me first rewrite this integration al's we can take three outside this integration so it would become three integration zero to pi over three. Sign cube dx. Sorry sine cubed X dx. Now if you refer the table or you could see the appendix D. That is given in the book we can use one formula which is integration. Sine cubed you do you as he calls you negative cause I knew plus 1/3 dozen cube you by using this formula we can easily evaluate this integration. Now if you reform sorry if you compare this integration with the given integral we can say that you as equals two X. And do you as equals two dx. So three integration zero to pi over three. Sign cube X. DX will be equals two negative. Cosine. You soak assign you we have ex Plus 1/3. Because in cuba So it would be co sign cube X. And the limit is from 0 to buy over three. Now this three would remain as it is. Now we're gonna put the limits upper limit minus a little of it. So three times Upper limit. We of course signed by over three Plus 1/3. This is because sign cube X. S by over a tree minus the lower limit. So lower limit we have zero because zero is one and this negative means negative one. Plus this is also causing cube zero is also one. So this would be 1/3. And because signed by over three we know it is 1/2. So it would be we can write this field this is 1/2 and cosine cubed by over three would be won over two and X cube, so it would become one over to an it's cube. Now we can evaluate this integral authorities uh expression and finally we'll get the answer to be three times. This is 5/24 and will get this value to be 5/8. Therefore, finally we can conclude that the definite integral which is zero to pi over three three sign cube DX is equal to 5/8. So this is a required answer. I hope you have understood the problem. Thank you.

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