Evaluate the indefinite integral: \int \sec^{2}x\tan^{4}xdx

Evaluate the indefinite integral:
$\int {\mathrm{sec}}^{2}x{\mathrm{tan}}^{4}xdx$
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Symbee
Step 1
Evaluate the indefinite integral.
$\int {\mathrm{sec}}^{2}x{\mathrm{tan}}^{4}xdx$
Let $\mathrm{tan}\left(x\right)=t$
${\mathrm{sec}}^{2}\left(x\right)dx=dt$
Step 2
$\int {\mathrm{sec}}^{2}\left(x\right){\mathrm{tan}}^{4}\left(x\right)dx=\int {t}^{4}dt$
$=\frac{{t}^{5}}{5}+c$
$=\frac{{\mathrm{tan}}^{5}\left(x\right)}{5}+c$
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Mary Ramirez
Step 1: Use Integration by Substitution.
Let $u=\mathrm{tan}x,du={\mathrm{sec}}^{2}xdx$
Step 2: Using u and du above, rewrite $\int {\mathrm{sec}}^{2}x{\mathrm{tan}}^{4}xdx$.
$\int {u}^{4}du$
Step 3: Use Power Rule: $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$.
$\frac{{u}^{5}}{5}$
Step 4: Substitute $u=\mathrm{tan}x$ back into the original integral.
$\frac{{\mathrm{tan}}^{5}x}{5}$
$\frac{{\mathrm{tan}}^{5}x}{5}+C$