Evaluate the following indefinite integral.
\[
\int 15 x^{\frac{1}{2}} d x
\]
Answer
The indefinite integral of \(15x^{\frac{1}{2}}\) with respect to \(x\) is \(\boxed{10x^{\frac{3}{2}} + C}\), where \(C\) is the constant of integration.
Step 1 ：The integral of a function in the form of \(x^n\) is given by \(\frac{x^{n+1}}{n+1}\). In this case, the function is \(15x^{\frac{1}{2}}\), so we can apply this rule to find the integral.
Step 2 ：The integral of the function \(15x^{\frac{1}{2}}\) is \(10x^{\frac{3}{2}}\). This is the antiderivative of the function, and represents the area under the curve of the function from 0 to x.
Step 3 ：The indefinite integral of \(15x^{\frac{1}{2}}\) with respect to \(x\) is \(\boxed{10x^{\frac{3}{2}} + C}\), where \(C\) is the constant of integration.
