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I am trying to find the integral of $$\int \tan x \sec^3 x dx$$

$$\int \tan x(1+\tan^2 x)\sec x\, dx$$

This gets me nowhere since I get a $\sec^2 x$ derivative with tan substitution so I try something else.

$$\int \frac {\sin x}{\cos x} \frac{1}{\cos^3x} dx$$

$$\int \frac {\sin x}{\cos^4 x} dx$$

$u = \cos x$ $du = -\sin x$

$$\int \frac {-1}{u^4} du$$

$$-1\int {u^{-4}} du$$

$$-1 \frac{u^{-3}}{3}$$ $$\frac{-1}{3\cos^3 x}$$

This for some reason is wrong.

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  • $\begingroup$ Your calculation is correct except for the last two lines. The integration of $-u^{-4}$ is $u^{-3}/3$, not $-u^{-3}/3$. $\endgroup$ – Sangchul Lee Jun 3 '12 at 17:24
  • $\begingroup$ You just missed a minus at the end. $\endgroup$ – Artem Jun 3 '12 at 17:24
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    $\begingroup$ (and for all of those constant-fans - add a +C) $\endgroup$ – H. M. Šiljak Jun 3 '12 at 17:26
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    $\begingroup$ Another method would be to note that $\tan x \, \sec^3 x = (\tan x \, \sec x)(\sec^2 x)$ and that the differential of $\sec x$ is $\tan x \, \sec x$. A method called spotting the presence of the derivative. $\endgroup$ – user26069 Jun 3 '12 at 17:30
  • $\begingroup$ Detailed steps here. $\endgroup$ – ablmf Dec 13 '19 at 21:19
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You missed a $-$ sign in the denominator. $$\int x^{n} \ \text dx = \frac{x^{n+1}}{n+1} + C$$ when $n \neq -1$.

So when $n=-4$ you get $$\int x^{-4} \ \text dx = -\frac{1}{3} \cdot x^{-3} +C$$

There is still an easier method of doing this: $$\int \tan{x}\cdot \sec^{3}(x) \ \text dx = \int \tan{x} \cdot \sec{x} \cdot \sec^{2}(x) \ \text dx = \int t^{2} \ \text dt$$ by putting $t=\sec{x}$.

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  • $\begingroup$ to the downvoter why a downvote $\endgroup$ – user9413 Jun 3 '12 at 17:34
  • $\begingroup$ @jordan: Great that you were able to come up with a solution on your own. Here is an exercise : $\int\frac{1}{\sqrt{\sin{x} \cdot \cos{x}}} \ dx$ $\endgroup$ – user9413 Jun 3 '12 at 17:45
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$$\int \frac{\sin(x)dx}{(\cos(x))^4}=-\int\frac{d(\cos(x))}{(\cos(x))^4}=\frac{1}{3(\cos(x))^3}+ C $$

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