Evaluate the Integral

Problem

(∫_^)(1/((x−2)(3/2))*d(x))

Solution

1. Rewrite the integrand using a negative exponent to make it easier to apply the power rule for integration.

(∫_^)((x−2)(−3/2)*d(x))

2. Apply the power rule for integration, which states that (∫_^)(un*d(u))=(u(n+1))/(n+1)+C for n≠−1

((x−2)(−3/2+1))/(−3/2+1)+C

3. Simplify the exponent and the denominator.

((x−2)(−1/2))/(−1/2)+C

4. Multiply by the reciprocal of the denominator and rewrite the negative exponent as a square root in the denominator.

−2*(x−2)(−1/2)+C

−2/√(,x−2)+C

Final Answer

(∫_^)(1/((x−2)(3/2))*d(x))=−2/√(,x−2)+C

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