Question

Integrate this following seriously hard equation urgh! The equation is
(Tanx^3 + tanx) / (2 secx^2 + tanx - 5) dx
Please

Answer 1

Is it \((tanx)^3\) or \(tan(x^3)\)? Same question for secx^2..

Answer 2

Its like the first one

Answer 3

OK! May I assist you?

Answer 4

\(\large \int \frac{tan x (1+tan^2x)}{2(1+tan^2x)+tan x -5}dx=\int \frac{tan x (sec^2x)}{2(tan^2x)+tan x -3}dx \)

Answer 5

@hartnn exactly! but you're just giving her the solution rather then helping her. How is she supposed to do other similar problems then?

Answer 6

i am sorry

Answer 7

only one step, ok

Answer 8

Thank you! Now would you @hartnn like to teach her to arrive at the correct solution or may I? It would help if you were to delete the answers you already wrote, although it's probably too late now sine she already saw them. Man! teach. Don't show off!

Answer 9

Yes yes im here! Please do!

Answer 10

now i won't give solution

Answer 11

OK @fatinatikah did you see any part of @hartnn 's solution?

Answer 12

Thank you @hartnn I appreciate that!

Answer 13

I only saw the first one. Are there more?

Answer 14

That's ok, but did you understand what @hartnn did because he is smart. He was right.

Answer 15

Okay i do. So for the next step wat technique should i use then?

Answer 16

@fatinatikah are you there. If you are could you please reply to my question. Did you understand what he did?

Answer 17

Yes yes i do. I want to kmow the next step. How do u integrate it then?

Answer 18

Not right away! First we need to simplify the fraction a bit. Could you show me what you think you saw as the first step. Then we'll take it from there.

Answer 19

If you understood it then you should be able to rewrite it, correct?

Answer 20

The first step is using the identity tan x (tan^2 x + 1) and also for the other one right?

Answer 21

Yes that was the numerator. Now what is the denominator?

Answer 22

So it will be tan x sec^2 x for the numerator right?

Answer 23

sec²x in the denominator reminds you of which Pythagorean trigonometric identity?

Answer 24

Then for the denominator is 2 tan^2 x + tan x - 5

Answer 25

Yes the numerator is correct. Now can you tell me the correct denominator for that numerator in that step? Please.

Answer 26

There is something missing. Can you figure it out? Do you know what it is?

Answer 27

What does sec²x equal? Think Pythagorean trigonometric identity.

Answer 28

Wait. Im sorry. I dont know. Im trying to figure it out

Answer 29

What are the three Pythagorean trigonometric identities? Do you know?

Answer 30

Sorry, take your time.

Answer 31

Is it tan^2 x. + 1 = sec^2 x ?

Answer 32

Perfect! So if you replace the sec²x in the denominator with it's identity, how does the denominator simplify? Work it out with pencil and pare and then write it here. Take your time.

Answer 33

*paper.

Answer 34

So do u find the factor of 2 tan^2 x + 5 tan x -3 ?

Answer 35

Not yet and where do you get 5tan x as the middle term? That's incorrect. Please try again.

Answer 36

Im sorry its 2 tan^2 x + tan x -3

Answer 37

Yes!

Answer 38

Now you have\[\int\limits_{}^{}\frac{ \tan x \sec ^{2}x }{ 2\tan ^{2}x +\tan x -3 }dx\]Correct?

Answer 39

Agree so far?

Answer 40

Yes i agree to that

Answer 41

Good! Now are you familiar with the substitution method?

Answer 42

If you are then what should you let u equal?

Answer 43

Okay so now we're using the substituition method? I was thinking of the partial fraction method

Answer 44

Not yet! Just work with me here, alright?

Answer 45

So what should u equal?

Answer 46

Okay. U = tan x is it?

Answer 47

Perfect!! Now if u = tan x then du = ?

Answer 48

Sec^2 x as the du

Answer 49

Again Perfect!! You're on a roll. Let's not loose this momentum now. OK so now after substitution, what will the integral look like?

Answer 50

Integral will look like u / (2u^2 + u -3) du

Answer 51

Beautiful! Now what do you propose we should do?

Answer 52

The partial fraction method?

Answer 53

Except you forgot the little integral sign in front. lol

Answer 54

Not yet!

Answer 55

Haha sorry. Okay i'll include that in my working later

Answer 56

Before we consider partial fractions, what should we do first.

Answer 57

Find the factor of 2u^2 + u -3

Answer 58

Correct! Factor the denominator. Go ahead and tell me the factored form of that polynomial.

Answer 59

(X-1) and (3x+2)

Answer 60

No, try again. If you expand or FOIL will you get the polynomial expression you tried to factor? I don't think so.

Answer 61

Please try again. I'm sure you probably made a careless error somewhere. We all do that so nothing to get discouraged over. Simply try again. Go ahead.

Answer 62

Alright sorry. Its u-1 and 2u+3

Answer 63

Excellent! Now what?

Answer 64

Now i'll break it into the a and b of the partial fraction?

Answer 65

Finally the partial fractions you've been wishing for all along. Great! So go ahead and tell me what you will get as your partial fractions. Take your time.

Answer 66

Hahah okay thanks. I think i get it from here. So its gonna be 3/2 ln (2u+3) + 1/5 ln (x-1) then substitute u back with tan x

Answer 67

You mean ln (u - 1). Please be careful!!!

Answer 68

Okay okay yes that

Answer 69

Now what do we do for the grand finale?

Answer 70

Substitute the u back into tan x

Answer 71

You mean replace each u with tan x and also don't forget to add a constant c at the end because without it the solution is still incorrect. Alright! So please let me see your final solution.

Answer 72

3/2 ln (tanx +3) + 1/5 ln (tan x -1) + c

Answer 73

Not quite! Almost but not quite! Please double check.

Answer 74

Oh wat is wrong then?

Answer 75

Also I just realized that before you had (3/2) ln(2u + 3) as the integral but in fact it's
(3/5)[(1/2) ln (2u + 3) which makes sense because the derivative of (1/2) ln (2u + 3) is
1/(2u + 3). Correct? Can you see where you went wrong?

Answer 76

You never actually told me what you got for A and B.

Answer 77

I should have asked you for the values of A and B before we went any further. Could you please tell me now?

Answer 78

Hint: Values for A and B are both rational numbers. What did you get?

Answer 79

Did she leave?

Answer 80

yup. she'll return , ithink, some net problems....

Answer 81

OH! Thanks!

Answer 82

Can you please tell me when she returns? Meanwhile I can perhaps help someone else. Thanks!

Answer 83

if i m online then i'll tell.

Answer 84

Thanks, actually I should get the notification also.

Answer 85

OK I see you're back.

Answer 86

Yes sorry net problem. Got hang. I got a and b as a fraction is that correct?

Answer 87

So Can you tell me now what you got for A and B?

Answer 88

3/2 and 1/5

Answer 89

I assume you mean A = 3/2 and B = 1/5. Only one of them is correct.

Answer 90

Please go back double check your work.

Answer 91

oh really? is it 2/3 then?

Answer 92

Which one do you think is incorrect and no, neither of them are 2/3. This is why I said that if you show me all your steps then I can see where you may have gone wrong.

Answer 93

Fine I'll make it a little easy for you. B = 1/5 is correct. A is wrong.

Answer 94

okay wait hold on ill figure it out

Answer 95

Take your time. I just want you to understand and be able to solve future problems with confidence.

Answer 96

okay thanks. @calculusfunctions i need to go now. i hope u'll be there for me if i have further questions needed to be asked. i perfectly understand it now. thanks!

Answer 97

For you reference before you go, you should have A = 3/5 and B= 1/5. And the final answer should be\[\frac{ 3 }{ 10 }\ln (2\tan x +3)+\frac{ 1 }{ 5 }\ln (\tan x -1)+c\]Otherwise, other than that careless mistake,you did great!