Question

evaluate the following indefinite integral.

∫10x^(2/3)dx

∫10x^(2/3)dx=□

Explanation:

Step1: Apply the power rule for integration

The power rule for integration is \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). 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:

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