Find the Antiderivative x square root of x

Problem

(∫_^)(x√(,x)*d(x))

Solution

1. Rewrite the expression using exponent notation by converting the square root to a power of one-half.

x√(,x)=x1⋅x(1/2)

2. Simplify the integrand by adding the exponents according to the product rule for powers.

x1⋅x(1/2)=x(3/2)

3. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1)+C for n≠−1

(∫_^)(x(3/2)*d(x))=(x(3/2+1))/(3/2+1)+C

4. Simplify the resulting fraction and exponent.

(x(5/2))/(5/2)+C=2/5*x(5/2)+C

Final Answer

(∫_^)(x√(,x)*d(x))=2/5*x(5/2)+C

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