"Evaluate the integral. ∫(x6−7x)4dx" solved and explained

High School
Calculus and Analysis
Calculus 2
Applications of integrals

osteoblogda

Answered question

2021-12-30

Evaluate the integral.

\int {(x^{6} - 7 x)}^{4} d x

Answer & Explanation

Thomas Lynn

Beginner2021-12-31Added 28 answers

Step 1
Consider the integral
$\int {(x^{6} - 7 x)}^{4} d x$
Step 2
let $I = \int {(x^{6} - 7 x)}^{4} d x$
remove x from the bracket
$I = \int x^{4} {(x^{5} - 7)}^{4} d x$
put $u = x^{5} - 7$
differentiating it wrt x we have
$d u = 5 x^{4} d x$
$\frac{d u}{5} = x^{4} d x$
Replace it with the integral
$I = \int u^{4} \frac{d u}{5}$
$I = \frac{1}{5} \int u^{4} d u$
$I = \frac{1}{5} (\frac{u^{5}}{5}) + c$
$I = \frac{{(x^{5} - 7)}^{5}}{25} + c$

Foreckije

Beginner2022-01-01Added 32 answers

\int {(x^{6} - 7 x)}^{4} d x

\int x^{24} - 28 x^{19} + 294 x^{14} - 1372 x^{9} + 2401 x^{4} d x

\int x^{24} d x - \int 28 x^{19} d x + \int 294 x^{14} d x - \int 1372 x^{9} d x + \int 2401 x^{4} d x

\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

Expert2022-01-04Added 613 answers

$\begin{matrix} \int (x^{6} - 7 x)^{4} d x \\ = \frac{1}{5} \int (u - 7)^{4} d u \\ \int (u - 7)^{4} d u \\ = \int v^{4} d v \\ = \frac{v^{5}}{5} \\ = \frac{(u - 7)^{5}}{5} \\ \frac{1}{5} \int (u - 7)^{4} d u \\ = \frac{(u - 7)^{5}}{25} \\ = \frac{(x^{5} - 7)^{5}}{25} \\ \int (x^{6} - 7 x)^{4} d x \\ = \frac{(x^{5} - 7)^{5}}{25} + C \end{matrix}$

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