This problem is relatively straight forward, but for some reason, my answer is off by the power of 1.

$$\int \tan \theta \sec^5\theta d\theta $$

The steps I take are

  • Step 1. $$ u = \sec \theta $$ $$ du = \tan\theta $$
  • Step 2. $$ \int u^5 du $$
  • Step 3. $$ (u^6 / 6) $$
  • Step 4. $$ \frac{(\sec\theta)^6}{6} + c $$

However, the answer according to wolfram is $$ \frac{(\sec\theta)^5}{5} + c $$

  • $\begingroup$ $du = sec \theta tan \theta$ $\endgroup$ – Don Larynx Sep 23 '13 at 1:00
  • $\begingroup$ $du = \sec\theta \tan\theta d\theta$. $\endgroup$ – Tunococ Sep 23 '13 at 1:00
  • $\begingroup$ Wow, can't believe I missed that. Thank you! $\endgroup$ – ConfusingCalc Sep 23 '13 at 1:00
  • $\begingroup$ You also happen to be off on your sentences by a word, as well. $\endgroup$ – Don Larynx Sep 23 '13 at 1:01
  • 1
    $\begingroup$ Erg, I changed the title and accidentally removed one word too many. Haha knock a guy while he is down. $\endgroup$ – ConfusingCalc Sep 23 '13 at 1:03

$$ \int \tan\theta\sec^5\theta\,d\theta = \int (\sec^4\theta)\Big( \tan\theta\sec\theta\, d\theta\Big) = \int u^4\, du. $$


You can also write the integral in terms of sines and cosines. You will then see that it is essentially a function in terms of cosines, since the sine is accounted for by the chain rule. A U-sub is overkill here. Bring the cosine term "up" and apply power rule for integrating.

  • $\begingroup$ Interesting way to look at it. Are you suggesting $ \int \frac{sinx}{cosx} * \frac{1}{cos^5(x)} $ then $ \int sinx * (cos^5(x))^{-1} $? $\endgroup$ – ConfusingCalc Sep 23 '13 at 2:08
  • $\begingroup$ How is it an overkill when OP substituted $u=\sec\theta$ and not an overkill when you choose to substitute $u=\sin\theta$? $\endgroup$ – peterwhy Sep 23 '13 at 2:08
  • $\begingroup$ $\displaystyle\int \frac{1}{\cos^6 x}\Big(\sin x\,dx\Big)$ $\displaystyle=\int\frac{1}{u^6}\,(-du)$. ${}\qquad{}$ $\endgroup$ – Michael Hardy Sep 23 '13 at 2:16
  • $\begingroup$ Peter, as Michael indicated, I am not using a substitution at all. The derivative of a cosine is a sine and thus I integrate the cosine function as if it were an x. Now, there is nothing wrong with substitution, I want to make that clear. But beyond Calc1, particularly in calc 2 and DFQ, we expect students to "see" the anti derivative if the chain rule is accounted for in an example worked out as I did above $\endgroup$ – imranfat Sep 23 '13 at 15:10

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