Question

Integrate the following w.r.t x 1) 2x^2/3.
2) (5-x)^23
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Answer 1

A)
`\int(2x^2)/(3)dx=(2x^3)/(9)+C`

B)
`\int(5-x)^(23)dx=-((5-x)^(24))/(24)+C`

Explanation:

A)

So we have the integral:

`\int(2x^2)/(3)dx`

First, remove the constant multiple:

`=(2)/(3)\int x^2\dx`

Use the power rule, where:

`\int x^ndx=(x^(n+1))/(x+1)`

Therefore:

`(2)/(3)\int x^2\dx\\=(2)/(3)((x^(2+1))/(2+1))`

Simplify:

`=(2)/(3)((x^(3))/(3))`

And multiply:

`=(2x^3)/(9)`

And, finally, plus C:

`=(2x^3)/(9)+C`

B)

We have the integral:

`\int(5-x)^(23)dx`

To solve, we can use u-substitute.

Let u equal 5-x. Then:

`u=5-x\\du=-1dx`

So:

`\int(5-x)^(23)dx\\=\int-u^(23)du`

Move the negative outside:

`=-\int u^(23)du`

Power rule:

`=-((u^(23+1))/(23+1))`

Add:

`=-((u^(24))/(24))`

Substitute back 5-x:

`=-(((5-x)^(24))/(24))`

Constant of integration:

`=-((5-x)^(24))/(24)+C`

And we're done!