Evaluate the Integral

Problem

(∫_^)(√(5,x3)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(5,x3)=x(3/5)

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

(∫_^)(x(3/5)*d(x))=(x(3/5+1))/(3/5+1)+C

3. Simplify the exponent and the denominator by adding the fractions.

3/5+1=8/5

4. Divide by the fraction by multiplying by its reciprocal.

(x(8/5))/(8/5)=5/8*x(8/5)

Final Answer

(∫_^)(√(5,x3)*d(x))=5/8*x(8/5)+C

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