Notes: `int sec^6(4x)tan(4x) dx` Find the indefinite integral

Given to solve,

$\displaystyle\int{{\sec}}^{{6}}{\left({4}{x}\right)}{\tan{{\left({4}{x}\right)}}}{\left.{d}{x}\right.}$

let $\displaystyle{u}={4}{x},{d}{u}={4}{\left.{d}{x}\right.}$

so,

$\displaystyle\int{{\sec}}^{{6}}{\left({4}{x}\right)}{\tan{{\left({4}{x}\right)}}}{\left.{d}{x}\right.}$

= $\displaystyle\int{{\sec}}^{{6}}{\left({u}\right)}{\tan{{\left({u}\right)}}}\frac{{{d}{u}}}{{4}}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\sec}}^{{6}}{\left({u}\right)}{\tan{{\left({u}\right)}}}{d}{u}$

let $\displaystyle{\sec{{u}}}={v}$

so, $\displaystyle{d}{v}={\left({\sec{{u}}}\right)}{\left({\tan{{u}}}\right)}{d}{u}$

so ,

$\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\sec}}^{{6}}{\left({u}\right)}{\tan{{\left({u}\right)}}}{d}{u}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\sec}}^{{5}}{\left({u}\right)}{\tan{{\left({u}\right)}}}\cdot{\sec{{\left({u}\right)}}}{d}{u}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\left({v}\right)}}^{{5}}{d}{v}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\frac{{{v}}^{{6}}}{{6}}$

= $\displaystyle\frac{{{{v}}^{{6}}}}{{24}}$

= $\displaystyle\frac{{{{\left({\sec{{\left({u}\right)}}}\right)}}^{{6}}}}{{24}}$

= $\displaystyle\frac{{{{\left({\sec{{\left({4}{x}\right)}}}\right)}}^{{6}}}}{{24}}+{c}$

Given to solve,

$\displaystyle\int{{\sec}}^{{6}}{\left({4}{x}\right)}{\tan{{\left({4}{x}\right)}}}{\left.{d}{x}\right.}$

let $\displaystyle{u}={4}{x},{d}{u}={4}{\left.{d}{x}\right.}$

so,

$\displaystyle\int{{\sec}}^{{6}}{\left({4}{x}\right)}{\tan{{\left({4}{x}\right)}}}{\left.{d}{x}\right.}$

= $\displaystyle\int{{\sec}}^{{6}}{\left({u}\right)}{\tan{{\left({u}\right)}}}\frac{{{d}{u}}}{{4}}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\sec}}^{{6}}{\left({u}\right)}{\tan{{\left({u}\right)}}}{d}{u}$

let $\displaystyle{\sec{{u}}}={v}$

so, $\displaystyle{d}{v}={\left({\sec{{u}}}\right)}{\left({\tan{{u}}}\right)}{d}{u}$

so ,

$\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\sec}}^{{6}}{\left({u}\right)}{\tan{{\left({u}\right)}}}{d}{u}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\sec}}^{{5}}{\left({u}\right)}{\tan{{\left({u}\right)}}}\cdot{\sec{{\left({u}\right)}}}{d}{u}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\int{{\left({v}\right)}}^{{5}}{d}{v}$

= $\displaystyle{\left(\frac{{1}}{{4}}\right)}\frac{{{v}}^{{6}}}{{6}}$

= $\displaystyle\frac{{{{v}}^{{6}}}}{{24}}$

= $\displaystyle\frac{{{{\left({\sec{{\left({u}\right)}}}\right)}}^{{6}}}}{{24}}$

= $\displaystyle\frac{{{{\left({\sec{{\left({4}{x}\right)}}}\right)}}^{{6}}}}{{24}}+{c}$

Notes