Evaluate the Integral

Problem

(∫_^)(y2√(,y)*d(y))

Solution

1. Rewrite the radical expression as a power with a fractional exponent.

√(,y)=y(1/2)

2. Simplify the integrand by using the product rule for exponents, adding the powers together.

y2⋅y(1/2)=y(2+1/2)

y(2+1/2)=y(5/2)

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

(∫_^)(y(5/2)*d(y))=(y(5/2+1))/(5/2+1)+C

4. Simplify the resulting fraction and exponent.

(y(7/2))/(7/2)+C

2/7*y(7/2)+C

Final Answer

(∫_^)(y2√(,y)*d(y))=2/7*y(7/2)+C

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