Guess and Check method integration

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mikewill54

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Hi,
Can someone help with this question, im using the guess and check method for integration from the calculus for dummies workbook

image_6487327 (1).JPG
But im not getting their answer when I differentiate [math]sec(5t-pi)[/math]
[math]sec(5T-pi)\\ \\Re-Write \ :sec(t)= \frac{1}{cosT} \\ =\frac{1}{cos(5T-pi)} \\ \underline{Use \ angle \ difference \ identity} \\Cos(s-t)=cos(s)cos(t)+sin(s)sin(t) \\=\frac{1}{Cos(5t)(Cos(pi)+Sin(5t)Sin(pi)} \\Cos(pi)=-1 \\Sin(pi)=0 \\=-\frac{1}{Cos(5t)} \\ \underline{Trig \ identity} \\ \frac{1}{cos(x)}=sec(x) \\So \\=\frac{d}{dx}(-sec(5x) \\-\frac{d}{dx}(sec(5x)) \\Use \ the \ chain \ rule \\f =sec(u) \ \ u=5x \\\frac{d}{du}sec(u) = sec(u)tan(u) \\\frac{d}{dx}=5 \\Substitute \ back \ in \\5.Sec(5x)tan(5x)[/math]
Thanks for any help
Regards
Mike
 
You seem to have dropped the minus.
Your answer is [imath]-5\sec{5x}\tan{5x}[/imath]
which is the same as [imath]5\sec(5x-\pi)\tan(5x-\pi)[/imath]

In any case, why not just use the chain rule on the original: [imath]\sec(5x-\pi)[/imath] rather than on [imath]-\sec{5x}[/imath]
 
Just what you did:
[imath]Use \ the \ chain \ rule \\f =sec(u) \ \ u=5x-\pi \\\frac{d}{du}sec(u) = sec(u)tan(u) \\\frac{du}{dx}=5 \\Substitute \ back \ in \\5.Sec(5x-\pi)tan(5x-\pi)[/imath]
 
The derivative of the sec of any angle is sec of that same angle times tangent of that some angle times the derivative of that angle. Repeat this until you got it.

So (sec(5x-π))' = sec(5x-π)tan(5x-π)*5
 
I think the guess and check method makes the problem more complicated than it needs to be.
If you recognize that [imath]\sec(5t-\pi)\tan(5t-\pi)=-\sec(5t)\tan(5t)[/imath], it can be solved with u-sub.
But I guess the point is to use the guess&check method.
 
Last edited:
To make live easier, I always let the angle be u (unless the angle is x or t...). That is I 2nd what BBB said
 
mikewill54 said:
Why not x or t?
Click to expand...
If you set u=x, then du = dx, then all you did was change the letter x to u. That is there was no gain in doing this. I advise you to do this one time and see for yourself what happens.
 
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