Evaluate the Integral

Problem

(∫_^)((x2−1)/(x3−3*x+1)*d(x))

Solution

1. Identify the structure of the integrand and notice that the numerator is related to the derivative of the denominator.

2. Define a substitution variable u for the denominator.

u=x3−3*x+1

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=3*x2−3

d(u)=3*(x2−1)*d(x)

4. Rearrange the differential to match the numerator of the integral.

1/3*d(u)=(x2−1)*d(x)

5. Substitute the expressions for u and d(u) into the integral.

(∫_^)(1/u⋅1/3*d(u))

6. Factor out the constant and integrate using the natural logarithm rule.

1/3*(∫_^)(1/u*d(u))

1/3*ln(u)+C

7. Back-substitute the original expression for u to get the final result.

1/3*ln(x3−3*x+1)+C

Final Answer

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

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