IMathAS-libretexts

Consider the integral

\displaystyle \int{x}^{{2}}{\left({x}^{{3}}-{5}\right)}^{{6}}{\left.{d}{x}\right.}

.

To find the value of this integral, make the substitution

\displaystyle {u}={x}^{{3}}-{5}

.

a) Write

\displaystyle {\left({x}^{{3}}-{5}\right)}^{{6}}

in terms of

\displaystyle {u}

:

b) This makes

\displaystyle {x}^{{2}}{\left.{d}{x}\right.}

\displaystyle {d}{u}

c) Re-write the integral in terms of

\displaystyle {u}

and

\displaystyle {d}{u}

, and find the antiderivative.
The antiderivative in terms of

\displaystyle {u}

is: + C

d) Now back-substitute and write your answer to part (c) in terms of

\displaystyle {x}

: + C