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Evaluate the indefinite integral given below.

int x(x-6)^(6)dx

Evaluate the indefinite integral given below. $\newline$ $\int x(x-6)^{6} d x$

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Find indefinite integrals using the substitution

Full solution

Q. Evaluate the indefinite integral given below.

\newline

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

Simplify Integral Expression: Step $1$ : Simplify the integral expression. $\newline$ We start by recognizing the integral as $\int x(x-6)^{6}\,dx$ .
Use Substitution: Step $2$ : Use substitution to simplify the integral. Let $u = x - 6$ , then $du = dx$ . When $x = 6$ , $u = 0$ . Substitute $x = u + 6$ into the integral. The integral becomes $\int (u+6)u^{6}\,du$ .
Expand and Integrate: Step $3$ : Expand the integrand and integrate term by term. $\newline$ $\int (u+6)u^{6}\,du = \int (u^{7} + 6u^{6})\,du$ $\newline$ = $\int u^{7}\,du + \int 6u^{6}\,du$ $\newline$ = $\frac{1}{8}u^{8} + \frac{6}{7}u^{7} + C$
Substitute Back for x: Step $4$ : Substitute back for x. $\newline$ Replace $u$ with $x - 6$ : $\newline$ $(\frac{1}{8})(x-6)^{8} + (\frac{6}{7})(x-6)^{7} + C$

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