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# Describe the technique for finding $\int \sec ^{5} x \tan ^{7} x dx$. Do not integrate.

Solution

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$\textbf{To solve}$ the given integral we will apply identity:

$\color{#c34632}{\tan^2{x}=\sec^2{x}-1}$

$\textbf{This leads}$ to:

\begin{align*} \int{\sec^5{x}\tan^7{x}\ dx}&=\int{\sec^4{x}(\sec^3{x}-1)^3\cdot \sec{x}\cdot \tan{x}\ dx}\\ \end{align*}

$\textbf{Now}$, we will apply the substitution:

$\color{#c34632}{u=\sec{x}\Rightarrow du=\sec{x}\tan{x}\ dx}$

$\textbf{Now}$, we have:

\begin{align*} \int{\sec^4{x}(\sec^3{x}-1)^3\cdot \sec{x}\cdot \tan{x}\ dx}&=\int{u^4(u^2-1)^3\ du}\\ \end{align*}

$\textbf{Last integral}$ is simple and we will not it solve since we are not asked to ...

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