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Struggling to integrate sec^2xtan^2x

Erotas

How to integrate sec^2xtan^2x.

I've tried it by parts but that doesn't seem to work.

Any help/hints much appreciated.

Reply 1

H.C. Chinaski

12

Original post

by Erotas

How to integrate sec^2xtan^2x.

I've tried it by parts but that doesn't seem to work.

Any help/hints much appreciated.

what is the derivative of tan(x)?

try a substitution u = tan(x)

with a bit of thought you can answer this mentally.

Reply 2

FireGarden

14

Sec and Tan have a very close relationship, in terms of trig identities. See if you can get it in terms of only tan or sec (or powers of) and see if that is easier to integrate.

Reply 3

H.C. Chinaski

12

Original post

by FireGarden

Sec and Tan have a very close relationship, in terms of trig identities. See if you can get it in terms of only tan or sec (or powers of) and see if that is easier to integrate.

that won't work.

Reply 4

Cupid93

Original post

by Erotas

How to integrate sec^2xtan^2x.

I've tried it by parts but that doesn't seem to work.

Any help/hints much appreciated.

Keep breaking it down until you find something you can work with:

Let u=sec^2 and v=tan^2
and if that's still too much at this stage:

Let a=sec b=sec c=tan d=tan
Differentiate a and b, use the product rule to find u; differentiate c and d, use the product rule to find v. Then just use the product rule on u and v.

Reply 5

H.C. Chinaski

12

Original post

by Cupid93

Keep breaking it down until you find something you can work with:

Let u=sec^2 and v=tan^2
and if that's still too much at this stage:

Let a=sec b=sec c=tan d=tan
Differentiate a and b, use the product rule to find u; differentiate c and d, use the product rule to find v. Then just use the product rule on u and v.

The question is on integration. the product rule has nothing to do with it.

the "method" is given in my first post.

Reply 6

SalBhatti

13

Use the formula $\displaystyle\int f^n(x) f^{'}(x)\ dx = \dfrac{f^{n+1}(x)}{n+1}$ to get $\dfrac{\tan^{3} x}{3} + k$ . Do you know how to do that?

Reply 7

steve2005

17

Original post

by H.C. Chinaski

The question is on integration. the product rule has nothing to do with it.

the "method" is given in my first post.

I think u = tan^2x is a better substitution

Reply 8

Danf159

2

Original post

by H.C. Chinaski

what is the derivative of tan(x)?

try a substitution u = tan(x)

with a bit of thought you can answer this mentally.

yup.

using substitution, u=tan (x) and du= sec^2 (x) dx

U can integrate normally from there and substitute tan(x) back in,
you'll get (1/3)tan^3 (x) + c

Reply 9

Femto

8

You are asked to evaluate:

$\displaystyle\int \sec^2 x\tan^2 x\ dx$

Let $u = \tan x$

$\implies du = \sec^2 x\ dx$

Carry out this substitution and evaluate the resulting integral.

Reply 10

Roy064

1

Original post

by Femto

You are asked to evaluate:

$\displaystyle\int \sec^2 x\tan^2 x\ dx$

Let $u = \tan x$

$\implies du = \sec^2 x\ dx$

Carry out this substitution and evaluate the resulting integral.

This is the one, beat me to the post.

Reply 11

H.C. Chinaski

12

Original post

by Danf159

yup.

using substitution, u=tan (x) and du= sec^2 (x) dx

U can integrate normally from there and substitute tan(x) back in,
you'll get (1/3)tan^3 (x) + c

indeed.

op

look out for situations where you are integrating f'(x)g(f(x))

all such cases can be solved with the substitution u=f(x)

(they can also be done by considering Integral g(f(x)) df(x))

Reply 12

H.C. Chinaski

12

Original post

by steve2005

I think u = tan^2x is a better substitution

how is that better????

with u=tan(x) you can see that the answer is (tan(x))^3 / 3 +c without any more thought????

with your approach you get u' = 2 tan(x) sec^2 (x) and I am afraid that this does not lead to an easier approach.

Reply 13

Average

isn't this simply a reverse chain rule....

Reply 14

Average

Original post

by Zishi

Use the formula $\displaystyle\int f^n(x) f^{'}(x)\ dx = \dfrac{f^{n+1}(x)}{n+1}$ to get $\dfrac{tan^{3} x}{3} + k$ . Do you know how to do that?

i think people ignore ur simple reverse chain rule lol ( y )

Reply 15

H.C. Chinaski

12

Original post

by Average

isn't this simply a reverse chain rule....

yes

Reply 16

Average

The easiest way to integrate tanx power n is to have sec^2x as it is the differentiation of tanx... All u need to know is chain rule.

Reply 17

H.C. Chinaski

12

Original post

by Average

i think people ignore ur simple reverse chain rule lol ( y )

I don't think that it is a case of ignoring rules as such, but rather that learning sets of rules that only work in specific cases is a methods approach rather than a mathematical approach and does not appeal to some people here. A-level text books are increasingly guilty of dumbing down understanding and replacing it with rote methods of the type:

If you see A then the answer is B
If you see C then the answer is D
If you see ....etc ....

A more fundamental approach encourages understanding and independence.

(edited 14 years ago)

Reply 18

Erotas

OP

Thanks for all your help guys!

I tried it first using u = tan^2x like steve2005 said which worked.
I also tried it using u = tanx which worked out to be tan^3x/3 really easily too. Much simpler than I thought.

Reply 19

Average

Original post

by H.C. Chinaski

I don't think that it is a case of ignoring rules as such, but rather that learning sets of rules that only work in specific cases is a methods approach rather than a mathematical approach and does not appeal to some people here. A-level text books are increasingly guilty of dumbing down understanding and replacing it with rote methods of the type:

If you see A then the answer is B
If you see C then the answer is D
If you see ....etc ....

A more fundamental approach encourages understanding and independence.

right... I hate rules too. However, integration is not mathematical. Unlike other concepts, integration is formed based on reversing differentiation. You would understand if you think about it thoroughly. Moreover, i think a level basics help with further understanding.

Struggling to integrate sec^2xtan^2x

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