Question

Integrate x+1/sqrt(x). I know that the answer is 2/3 • sqrt(x) • (x+3) + C, but I don't know how it is simplified from 2/3 • x^3/2 + 2 • x^1/2 + C. Please explain this step in detail? Thank you!

Answer 1

To simplify the integration result, factor out the common term 2 · x^{1/2}, which is then recognized as the square root of x, resulting in the final expression 2/3 · sqrt(x) · (x+3) + C.

Step-by-step explanation:

You're asking how to simplify the expression 2/3 · x^{3/2} + 2 · x^{1/2} + C to 2/3 · sqrt(x) · (x+3) + C. I'll walk you through the steps.

First, let's factor out the common term from the first two terms, which is 2 · x^{1/2}:

x^{1/2} can be factored out because it is a common factor of both x^{3/2} and x^{1/2}.

• Express x^{3/2} as x^{1/2} · x and factor out x^{1/2}:
  2/3 · (x^{1/2} · x) + 2 · x^{1/2} becomes 2/3 · x^{1/2} · (x + 3).
• Now recognize that x^{1/2} is the square root of x, and rewrite the expression to the final simplified form: 2/3 · sqrt(x) · (x + 3) + C.

Answer 2

`\displaystyle\int(x+1)/(\sqrt x)\,\mathrm dx=\int\frac x{x^(1/2)}+\frac1{x^(1/2)}\,\mathrm dx`

`=\displaystyle\int x^(1/2)+x^(-1/2)\,\mathrm dx`

By the power rule,

`=(x^(3/2))/(\frac32)+(x^(1/2))/(\frac12)+C`

(this seems to be the step you're not getting?)

`=\frac23x^(3/2)+2x^(1/2)+C`

The next step is to pull out a common factor of
`x^(1/2)` from the antiderivative:

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

so that the final result is

`\frac23x^(3/2)+2x^(1/2)+C=\frac23x^(1/2)(x+3)+C`