### Video Transcript

Calculate the integral of π to the
π₯ power times one minus π to the negative π₯ power squared dπ₯.

The first thing that we can do here
is take this one minus π to the negative π₯ power squared and expand it. Weβre saying one minus π to the
negative π₯ power times itself. One minus π to the negative π₯
power. Bring down the π to the π₯ and the
dπ₯. Inside this integral, weβre
multiplying these three terms together. To simplify, we could multiply one
minus π to the negative π₯ times one minus π to the negative π₯. And then, whatever we got there, we
would multiply by π to the π₯ power.

However, because weβre only dealing
with multiplication, we can also multiply π to the π₯ power times one minus π to
the negative π₯ power. And then from there, weβll take
whatever we get and multiply that by one minus π to the negative π₯ power. Letβs do the second option.

Weβll distribute this π to the π₯
power over one minus π to the negative π₯, which is π to the π₯, minus π to the
π₯ power times π to the negative π₯ power. We have to be careful here. Remembering our power rules, if
weβre multiplying π to the π power times π to the π power, what we do is add the
exponents, π plus π. In this case, that means weβll be
adding π₯ plus negative π₯. π to the π₯ power times π to the
negative π₯ power equals π to the zero power, which equals one.

We need to multiply π to the π₯
power minus one times one minus π to the negative π₯ power. π to the π₯ power times one equals
π to the π₯. And now, we have π to the π₯ power
times negative π to the negative π₯ power. So it looks like this: minus π to
the power times π to the negative π₯ power, which we know is π to the zero
power. Then, we have negative one times
one which is negative one and negative one times negative π to the negative π₯
power which is positive π to the negative π₯ power. Bringing down what we know, π to
the π₯ power minus π to the zero β which is one β minus one plus π to the negative
π₯ power. One minus one equals negative
two.

What we have here π to the π₯
minus two plus π to the negative π₯ is a simplified expression of what we started
with. We can rewrite our original
integral to say we wanna find the integral of π to the π₯ power plus π to the
negative π₯ power minus two dπ₯. And this is much more
manageable. Weβll just find the integral of
each individual terms and add them together. The integral of π to the π₯ dπ₯
equals π to the π₯ power.

The integral of π to the negative
π₯ power with respect to π₯ is negative π to the negative π₯ power. And weβre subtracting the integral
of two with respect to π₯, which is negative two π₯ plus any constant π. The integral is π to the π₯ power
minus π to the negative π₯ power minus two π₯ plus π. The key here was to simplify the
expression that we were originally given into a format that was easier to
integrate.