Evaluate the Integral

Problem

(∫_^)(4√(3,x2)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(3,x2)=x(2/3)

2. Move the constant factor outside the integral.

(∫_^)(4*x(2/3)*d(x))=4*(∫_^)(x(2/3)*d(x))

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

n=2/3

n+1=2/3+1=5/3

4. Substitute the values into the power rule formula.

4⋅(x(5/3))/(5/3)+C

5. Simplify the expression by multiplying by the reciprocal of the denominator.

4⋅3/5*x(5/3)+C

12/5*x(5/3)+C

Final Answer

(∫_^)(4√(3,x2)*d(x))=(12*x(5/3))/5+C

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