evaluate the following indefinite integral.\n\n∫10x^(2/3)dx\n\n∫10x^(2/3)dx=□

Explanation:

Step1: Apply the power rule for integration

The power rule for integration is (\int x^n dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)). For the integral (\int10x^{\frac{2}{3}}dx), we can use the constant - multiple rule (\int kf(x)dx=k\int f(x)dx) (where (k = 10) and (f(x)=x^{\frac{2}{3}})). So, (\int10x^{\frac{2}{3}}dx=10\int x^{\frac{2}{3}}dx). Using the power rule with (n=\frac{2}{3}), we have (10\times\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1}+C).

Step2: Simplify the expression

First, simplify the exponent and the denominator. (\frac{2}{3}+1=\frac{2 + 3}{3}=\frac{5}{3}). Then, (10\times\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+C). When we divide by a fraction, we multiply by its reciprocal. So, (10\times\frac{3}{5}x^{\frac{5}{3}}+C). Since (10\times\frac{3}{5}=6), the integral (\int10x^{\frac{2}{3}}dx = 6x^{\frac{5}{3}}+C).

Answer:

(6x^{\frac{5}{3}}+C)