Evaluate the Integral

Problem

(∫_^)((x2)/(3−x3)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the numerator x2 is a multiple of the derivative of the denominator 3−x3

2. Substitute u=3−x3 to simplify the integrand.

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=−3*x2

d(u)=−3*x2*d(x)

4. Solve for the expression found in the numerator by dividing both sides by −3

−1/3*d(u)=x2*d(x)

5. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(1/u*(−1/3)*d(u))

−1/3*(∫_^)(1/u*d(u))

6. Integrate using the rule (∫_^)(1/u*d(u))=ln(u)+C

−1/3*ln(u)+C

7. Back-substitute the original expression 3−x3 for u to get the final result in terms of x

−1/3*ln(3−x3)+C

Final Answer

(∫_^)((x2)/(3−x3)*d(x))=−1/3*ln(3−x3)+C

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