Evaluate the Integral

Problem

(∫_^)(√(3,x)*(x−4)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(3,x)=x^(1/3)

2. Distribute the term x^(1/3) into the binomial (x−4)

x^(1/3)*(x−4)=x^(1/3)⋅x^1−4*x^(1/3)

3. Simplify the exponents using the rule x^a⋅x^b=x^(a+b)

x^(1/3+1)−4*x^(1/3)=x^(4/3)−4*x^(1/3)

4. Apply the power rule for integration, (∫_^)(x^n*d(x))=(x^(n+1))/(n+1) to each term.

(∫_^)(x^(4/3)*d(x))−(∫_^)(4*x^(1/3)*d(x))

5. Calculate the new exponents and their coefficients.

(x^(4/3+1))/(4/3+1)−4⋅(x^(1/3+1))/(1/3+1)

6. Simplify the fractions in the denominators and coefficients.

(x^(7/3))/(7/3)−4⋅(x^(4/3))/(4/3)

7. Multiply by the reciprocals and add the constant of integration C

3/7*x^(7/3)−4⋅3/4*x^(4/3)+C

8. Finalize the coefficients by multiplying.

3/7*x^(7/3)−3*x^(4/3)+C

Final Answer

(∫_^)(√(3,x)*(x−4)*d(x))=3/7*x^(7/3)−3*x^(4/3)+C

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