Evaluate the Integral

Problem

(∫_^)(√(3,x8)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(3,x8)=x(8/3)

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

(∫_^)(x(8/3)*d(x))=(x(8/3+1))/(8/3+1)+C

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

8/3+1=8/3+3/3=11/3

4. Divide by the new exponent, which is equivalent to multiplying by its reciprocal.

(x(11/3))/(11/3)=3/11*x(11/3)

Final Answer

(∫_^)(√(3,x8)*d(x))=(3*x(11/3))/11+C

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