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=>int(x+1)/(sqrtx)dx.

$\Rightarrow \int \frac{x+1}{\sqrt{x}} d x$ .

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Find indefinite integrals using the substitution and by parts

Full solution

Q.

\Rightarrow \int \frac{x+1}{\sqrt{x}} d x

.

Rewrite in terms of x: Rewrite the integral in terms of x to simplify the integration process. $\newline$ $I = \int(\frac{x+1}{\sqrt{x}}) dx$ $\newline$ $I = \int(\frac{x}{\sqrt{x}} + \frac{1}{\sqrt{x}}) dx$ $\newline$ $I = \int(x^{\frac{1}{2}} + x^{-\frac{1}{2}}) dx$
Integrate each term: Integrate each term separately. $\newline$ $I = \int x^{\frac{1}{2}} \, dx + \int x^{-\frac{1}{2}} \, dx$
Apply power rule: Apply the power rule for integration to each term. $\newline$ $I = \frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} + C$ , where $C$ is the constant of integration.
Check by differentiation: Check the result by differentiating it to see if we get the original integrand. $\newline$ $\frac{d}{dx}\left[\frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}}\right] = \frac{2}{3}\cdot\frac{3}{2}x^{\frac{1}{2}} + 2\cdot\frac{1}{2}x^{-\frac{1}{2}}$ $\newline$ $\frac{d}{dx}\left[\frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}}\right] = x^{\frac{1}{2}} + x^{-\frac{1}{2}}$ $\newline$ $\frac{d}{dx}\left[\frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}}\right] = \frac{x+1}{\sqrt{x}}$ $\newline$ The differentiation gives us the original integrand, so there is no math error.

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