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Integrate each of the given functions.$$\int_{0}^{4} \frac{9 d t}{\left(t^{2}+9\right)^{3 / 2}}$$

$\frac{4}{5}$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 8

Integration by Trigonometric Substitution

Integrals

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Calculate the integral.

So hello, have we gone today? We're gonna solve the integral. Comes here to throw it nine over T square plus nine. The whole listed three by two. Now this is a econometric substitution question. And so t would be if you notice this in the form of X square. Yes things clear the whole room and you get the Can you use the substitution of excessive from a 10 ft on this? So um X as they come to a 10 m. Mhm. In this equation he would be able to three and peter because the are Square root of nine is 3. So duty moving on. The UK would be the differentiation of this. But you can do three sex. Where do you draw now? Um Okay duty here. Uh So now I mean you just need to replace these equations back into the main problem. So that leaves 02. Or actually now you you can change your endpoints because you're going to take a reform theater. So they come from 02 People can be written as down and worse, three by three. So then it was zero is zero. And then this can be organized diamond wires. Sorry for the background knives. Yes. Women that Than was four x 3. Now writing this down, they can take the nine outside and this will give you one hour P square which is um 3 10%. 3 square is nine. So you can square theater plus nine for Asia three x 2 into DT which is trees. Sec square theater. Now um the in you can simplify this further, giving 02 and in worse four by three. Now You can take the nine common and you'll get one where brian Coleman one work and nine times square Utah plus one. A recently by two. I have reckoned trail here. That's that was my mistake. Mr three x 2 into into three fact square meter. No Uh transcript Pita Plus one gives you sex square theater. So do you go 10 and worse of four x 3. I need to read this in a better there. Yeah. Coming into my equation four by three. So now you can write this as 1/9. Okay. Paracetamol agent. Uh hft righto By two and 2 three. Check square meter. Um Now if you simplify this further, you would get uh, you can take the nine outside And get nine into 1/9. 02 integral from 0 to 10. And where's four by three. Um, one over sick peter peter. So what did you do that river duty to don't forget to write devi device. Um, Now, now that now you can, that's quite a, is the integral. So you can some feathers a bit further by writing. It has The nine Guns Cloud Time and worst of four x 3 integral. From 0 to 9 and verse. Four by three And now six computers. Seconds. 2nd data is one of our cars data and this is one of our sector. As a cost. People will come on the numerator side and this will just give you cross peter peter. The integral. Of course tita is signed to um cell phone. So from that will be signed tita from its usual too. Mark and four by three. Part time for Betty. And now If you put this in the calculator, you get the answer. S four x 5. So I hope you guys understood that. That's your final answer and thanks for watching.

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