osteoblogda

2021-12-30

Evaluate the integral.
$\int {\left({x}^{6}-7x\right)}^{4}dx$

Thomas Lynn

Step 1
Consider the integral

Step 2
let
remove x from the bracket

put $u={x}^{5}-7$
differentiating it wrt x we have

Replace it with the integral
$I=\int {u}^{4}\frac{du}{5}$
$I=\frac{1}{5}\int {u}^{4}du$
$I=\frac{1}{5}\left(\frac{{u}^{5}}{5}\right)+c$
$I=\frac{{\left({x}^{5}-7\right)}^{5}}{25}+c$

Foreckije

$\int {\left({x}^{6}-7x\right)}^{4}dx$
$\int {x}^{24}-28{x}^{19}+294{x}^{14}-1372{x}^{9}+2401{x}^{4}dx$
$\int {x}^{24}dx-\int 28{x}^{19}dx+\int 294{x}^{14}dx-\int 1372{x}^{9}dx+\int 2401{x}^{4}dx$
$\frac{{x}^{25}}{25}-\frac{7{x}^{20}}{5}+\frac{98{x}^{15}}{5}-\frac{686{x}^{10}}{5}+\frac{2401{x}^{5}}{5}$
$\frac{{x}^{25}}{25}+\frac{-7{x}^{20}+98{x}^{15}-686{x}^{10}+2401{x}^{5}}{5}$
$\frac{{x}^{25}}{25}+\frac{-7{x}^{20}+98{x}^{15}-686{x}^{10}+2401{x}^{5}}{5}+C$

karton

$\begin{array}{}\int \left({x}^{6}-7x{\right)}^{4}dx\\ =\frac{1}{5}\int \left(u-7{\right)}^{4}du\\ \int \left(u-7{\right)}^{4}du\\ =\int {v}^{4}dv\\ =\frac{{v}^{5}}{5}\\ =\frac{\left(u-7{\right)}^{5}}{5}\\ \frac{1}{5}\int \left(u-7{\right)}^{4}du\\ =\frac{\left(u-7{\right)}^{5}}{25}\\ =\frac{\left({x}^{5}-7{\right)}^{5}}{25}\\ \int \left({x}^{6}-7x{\right)}^{4}dx\\ =\frac{\left({x}^{5}-7{\right)}^{5}}{25}+C\end{array}$