Question

What is the antiderivative of `(16 - x)^(3/2)`?

Answer 1

`-2/5(16-x)^(3/2) + C`

Explanation:

Antiderivative is just the integration. For handling this type of problem, you should be comfortable with the power rule.

`color(red)(intquad x^n dx = x^(n+1)/(n+1)+C`

Our problem we have to find

`intquad(16-x)^(3/2) dx`

We shall use a `u` substitution just to make into a form which we are comfortable with.

Let `16-x = u`
Differentiating with respect to `x` on both the sides we get.

`-dx= du`
`dx=-du`

Our integral now becomes

`intquadu^(3/2)(-du)`
`=-intquad u^(3/2)du`

Use the power rule

`=-u^(3/2+1)/(3/2+1) +C`
`=-u^(5/2)/(5/2)+C`
`=-2/5u^(5/2)+C`

Substituting back for `u` we get,

`=-2/5(16-x)^(3/2) + C`