Question

Question #946b3

Answer 1

`int (x^{1/2} + 3x + 5) dx = 2/3 x^{3/2} + 3/2 x^2 + 5x + "constant"`

Explanation:

Use the power rule.

For `n != -1`,

`int x^n dx = x^{n+1}/{n+1} + "constant"`.

Therefore,

`int (x^{1/2} + 3x + 5) dx = int x^{1/2} dx + 3int x dx + 5int x^0 dx`

`= 2/3 x^{3/2} + 3/2 x^2 + 5x + "constant"`

Answer 2

`2/3x^(3/2)+3/2x^2+5x+c`

Explanation:

This is a Calculus question.

Integrate each term using the `color(blue)"power rule for integration"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(intax^ndx=a/(n+1)x^(n+1) ; n≠-1)color(white)(2/2)|)))`

`color(blue)"Note that " 5=5x^0" since " x^0=1`

`rArrint(x^(1/2)+3x+5)dx`

`=1/(3/2)x^((1/2+1))+3/2x^((1+1))+5x^((0+1))+c`

`=2/3x^(3/2)+3/2x^2+5x+c`

where c is the constant of integration.