Integrate integral ((1+x)²)/(sqrt(x)) dx

Question

ScanSolve AI Tutor · Accepted Answer

To integrate the given expression, we will first expand the numerator and then divide each term by the denominator.

Expand the numerator $(1+x)^2$ .
$(1+x)^2 = 1^2 + 2(1)(x) + x^2 = 1 + 2x + x^2$
Rewrite the integral with the expanded numerator and express $\sqrt{x}$ as $x^{1/2}$ .
$\int \frac{1 + 2x + x^2}{x^{1/2}}\,dx$
Divide each term in the numerator by $x^{1/2}$ .
$\int \left(\frac{1}{x^{1/2}} + \frac{2x}{x^{1/2}} + \frac{x^2}{x^{1/2}}\right)\,dx$ $\int \left(x^{-1/2} + 2x^{1 - 1/2} + x^{2 - 1/2}\right)\,dx$ $\int \left(x^{-1/2} + 2x^{1/2} + x^{3/2}\right)\,dx$
Integrate each term using the power rule for integration, $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ .
For the first term, $x^{-1/2}$ :
$\int x^{-1/2}\,dx = \frac{x^{-1/2+1}}{-1/2+1} = \frac{x^{1/2}}{1/2} = 2x^{1/2}$
For the second term, $2x^{1/2}$ :
$\int 2x^{1/2}\,dx = 2 \frac{x^{1/2+1}}{1/2+1} = 2 \frac{x^{3/2}}{3/2} = 2 \cdot \frac{2}{3} x^{3/2} = \frac{4}{3} x^{3/2}$
For the third term, $x^{3/2}$ :
$\int x^{3/2}\,dx = \frac{x^{3/2+1}}{3/2+1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}$
Combine the integrated terms and add the constant of integration, $C$ .
$2x^{1/2} + \frac{4}{3} x^{3/2} + \frac{2}{5} x^{5/2} + C$

Answer