# Evaluate the following integrals. \tan^{7}x \sec^{4}xdx

Evaluate the following integrals.
${\mathrm{tan}}^{7}x{\mathrm{sec}}^{4}xdx$
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Step 1
To Determine:
Evaluate the following integrals.
Given: we have
${\mathrm{tan}}^{7}x{\mathrm{sec}}^{4}xdx$
Explanation: we have
${\mathrm{tan}}^{7}x{\mathrm{sec}}^{4}xdx$
we can write
$\int {\mathrm{tan}}^{7}x{\mathrm{sec}}^{2}x{\mathrm{sec}}^{2}xdx$
$\int \left(1-{\mathrm{tan}}^{2}x\right){\mathrm{sec}}^{2}x{\mathrm{tan}}^{7}xdx$
substituting $u=\mathrm{tan}xdu={\mathrm{sec}}^{2}xdx$ then we have
$\int {u}^{7}\left(1+{u}^{2}\right)du=\int \left({u}^{7}+{u}^{9}\right)du$
Step 2
$\int \left({u}^{7}+{u}^{9}\right)du=\frac{{u}^{8}}{8}+\frac{{u}^{10}}{10}+c$
$\frac{{\mathrm{tan}}^{8}x}{8}+\frac{{\mathrm{tan}}^{10}x}{10}+c$