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Rewriting fractions

shamieh

Active member
Sep 13, 2013
539
How can I rewrite \(\displaystyle \int \frac{tan^3x}{cos^3x} \, dx\) to \(\displaystyle \int tan^3x sec^3x \, dx\)

What is the identity they are using to do this?
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492

shamieh

Active member
Sep 13, 2013
539

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
I am saying they used the definition of $\sec x$ to rewrite the first expression in post #1 to the second one. This is not related to integration; they rewrote purely the function being integrated.
 

shamieh

Active member
Sep 13, 2013
539
Yes, but \(\displaystyle secx = \frac{1}{cosx}\) not \(\displaystyle tan^3sec^3x\)

can you show me what's going on?

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even if you re wrote it you would still have tan^3x/sec^3x

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Because you don;t have 1/cosx you have tan^3x/cos^3x
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
even if you re wrote it you would still have tan^3x/sec^3x
Surely not...

$\displaystyle \begin{align*} \frac{\tan^3{(x)}}{\cos^3{(x)}} &= \tan^3{(x)} \left[ \frac{1}{\cos^3{(x)}} \right] \\ &= \tan^3{(x)} \left[ \frac{1}{\cos{(x)}} \right] ^3 \\ &= \tan^3{(x)} \sec^3{(x)} \end{align*}$
 

shamieh

Active member
Sep 13, 2013
539
Oh I see now.. I didn't know you could rewrite cos^3x to 1/cosx
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
Oh I see now.. I didn't know you could rewrite cos^3x to 1/cosx
You can't. But you CAN write $\displaystyle \begin{align*} \frac{1}{\cos^3{(x)}} \end{align*}$ as $\displaystyle \begin{align*} \left[ \frac{1}{\cos{(x)}} \right] ^3 \end{align*}$.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Somehow the following post did not show earlier. I must have accidentally closed it before posting.

By definition,
\[
\frac{1}{\cos x}=\sec x.\]
Taking the cube of both sides,
\[
\frac{1}{\cos^3 x}=\left(\frac{1}{\cos x}\right)^3=(\sec x)^3=\sec^3x.
\]
Multiplying both sides by $\tan^3 x$ we get
\[
\frac{\tan^3 x}{\cos^3 x}=\tan^3 (x)\sec^3x.
\]