Krzychau1

2022-01-14

Evaluate $\int {\mathrm{cos}}^{2}\left(x\right){\mathrm{tan}}^{3}\left(x\right)dx$ using trigonometric substitution

Vivian Soares

Expert

$\int {\mathrm{cos}}^{2}x{\mathrm{tan}}^{3}xdx=\int \frac{{\mathrm{sin}}^{3}x}{\mathrm{cos}x}dx=\int \frac{\left(1-{\mathrm{cos}}^{2}x\right)\mathrm{sin}x}{\mathrm{cos}x}dx$
Let $\mathrm{cos}x=t⇒-\mathrm{sin}xdx=dt$
$=\int \frac{\left({t}^{2}-1\right)dt}{t}$
$=\int \left(t-\frac{1}{t}\right)dt$

sonorous9n

Expert

Use the definition of $\mathrm{tan}x$, then after a bit of algebra you have
$\frac{\mathrm{sin}x}{\mathrm{cos}x}-\mathrm{cos}x\mathrm{sin}x$
The first one is solved using logarithm subsitution, the second using the identity $2\mathrm{sin}x\mathrm{cos}x=\mathrm{sin}2x$

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