\[\int \tan^{3}x \, dx\]
1
Use Pythagorean Identities: tan⁡2x=sec⁡2x−1\tan^{2}x=\sec^{2}x-1.
∫(sec⁡2x−1)tan⁡x dx\int (\sec^{2}x-1)\tan{x} \, dx

2
Expand.
∫tan⁡xsec⁡2x−tan⁡x dx\int \tan{x}\sec^{2}x-\tan{x} \, dx

3
Use Sum Rule: ∫f(x)+g(x) dx=∫f(x) dx+∫g(x) dx\int f(x)+g(x) \, dx=\int f(x) \, dx+\int g(x) \, dx.
∫tan⁡xsec⁡2x dx−∫tan⁡x dx\int \tan{x}\sec^{2}x \, dx-\int \tan{x} \, dx

4
Use Integration by Substitution on ∫tan⁡xsec⁡2x dx\int \tan{x}\sec^{2}x \, dx.
Let u=tan⁡xu=\tan{x}, du=sec⁡2x dxdu=\sec^{2}x \, dx

5
Using uu and dudu above, rewrite ∫tan⁡xsec⁡2x dx\int \tan{x}\sec^{2}x \, dx.
∫u du\int u \, du

6
Use Power Rule: ∫xn dx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
u22\frac{{u}^{2}}{2}

7
Substitute u=tan⁡xu=\tan{x} back into the original integral.
tan⁡2x2\frac{\tan^{2}x}{2}

8
Rewrite the integral with the completed substitution.
tan⁡2x2−∫tan⁡x dx\frac{\tan^{2}x}{2}-\int \tan{x} \, dx

9
Use Trigonometric Integration: the integral of tan⁡x\tan{x} is ln⁡(sec⁡x)\ln{(\sec{x})}.
tan⁡2x2−ln⁡(sec⁡x)\frac{\tan^{2}x}{2}-\ln{(\sec{x})}

10
Add constant.
tan⁡2x2−ln⁡(sec⁡x)+C\frac{\tan^{2}x}{2}-\ln{(\sec{x})}+C

Done
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