Evaluate the Integral integral of 1/(x^2-2x) with respect to x

Problem

(∫_^)(1/(x2−2*x)*d(x))

Solution

1. Factor the denominator to prepare for partial fraction decomposition.

(∫_^)(1/(x*(x−2))*d(x))

2. Set up the partial fraction decomposition by splitting the integrand into two simpler fractions.

1/(x*(x−2))=A/x+B/(x−2)

3. Solve for the constants A and B by multiplying both sides by x*(x−2)

1=A*(x−2)+B*x

4. Substitute x=0 to find A

1=A*(0−2)⇒A=−1/2

5. Substitute x=2 to find B

1=B(2)⇒B=1/2

6. Rewrite the integral using the partial fractions found.

(∫_^)(((−1/2)/x+(1/2)/(x−2))*d(x))

7. Integrate each term separately using the natural logarithm rule.

−1/2*ln(x)+1/2*ln(x−2)+C

8. Simplify the expression using the properties of logarithms.

1/2*ln((x−2)/x)+C

Final Answer

(∫_^)(1/(x2−2*x)*d(x))=1/2*ln((x−2)/x)+C

Want more problems? Check here!