Evaluate the Integral

Problem

(∫_^)(4*x3−3/(x3)*d(x))

Solution

1. Rewrite the integrand using negative exponents to make it easier to apply the power rule.

(∫_^)(4*x3−3*x(−3)*d(x))

2. Apply the sum rule for integration to split the expression into two separate integrals.

(∫_^)(4*x3*d(x))−(∫_^)(3*x(−3)*d(x))

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

(4*x(3+1))/(3+1)−(3*x(−3+1))/(−3+1)+C

4. Simplify the fractions and the exponents.

(4*x4)/4−(3*x(−2))/(−2)+C

5. Simplify the coefficients and rewrite the negative exponent as a fraction.

x4+3/(2*x2)+C

Final Answer

(∫_^)(4*x3−3/(x3)*d(x))=x4+3/(2*x2)+C

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