Evaluate the Integral

Problem

(∫_^)(x√(,x)*d(x))

Solution

1. Rewrite the integrand using exponent notation by expressing the square root as a power of 1/2

(∫_^)(x⋅x(1/2)*d(x))

2. Combine the terms by adding the exponents, since x1⋅x(1/2)=x(1+1/2)

(∫_^)(x(3/2)*d(x))

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+1))/(3/2+1)+C

4. Simplify the exponent and the denominator.

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

5. Multiply by the reciprocal of the denominator to find the final form.

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

Final Answer

(∫_^)(x√(,x)*d(x))=2/5*x(5/2)+C

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