Question

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.

\(\int {{\rm{x}}\sqrt {{\rm{xdx}}} } \)

Short answer

The value of the integral is \(\frac{{\rm{2}}}{{\rm{5}}}{{\rm{x}}^{{{\rm{5}} \mathord{\left/

{\vphantom {{\rm{5}} {}}} \right.

\kern-\nulldelimiterspace} {}}{\rm{2}}}}{\rm{ + C}}\)

Step-by-step solution

Evaluating the indefinite integral.

\(\begin{aligned}{c}\int {\rm{x}} \sqrt {\rm{x}} {\rm{dx = }}\int {\rm{x}} \left( {{{\rm{x}}^{\frac{{\rm{1}}}{{\rm{2}}}}}} \right){\rm{dx}}\\{\rm{ = }}\int {{{\rm{x}}^{{\rm{1 + }}\frac{{\rm{1}}}{{\rm{2}}}}}} {\rm{dx = }}\int {{{\rm{x}}^{\frac{{\rm{3}}}{{\rm{2}}}}}} {\rm{dx}}\\{\rm{ = }}\frac{{\rm{2}}}{{\rm{5}}}{{\rm{x}}^{{\rm{5/2}}}}{\rm{ + C}}\end{aligned}\)

Graph of the integral.

The graph of integral with different values of constants is shown in the figure below.