To evaluate the integral, we'll complete the square in the denominator and use a trigonometric substitution. Here's the breakdown:
1. Completing the Square:
The denominator is
2x−x2. To complete the square, we factor out a -1 and rearrange:
2x−x2=−(x2−2x)=−(x2−2x+1−1)=−(x−1)2+1
2. Trigonometric Substitution:
Now, we use the substitution
x−1=sin‌θ, which implies
dx=cos‌θ‌d‌θ. Also,
(x−1)2=sin‌2θ. Substituting into the integral, we get:
3. Simplifying and Integrating:
Using the identity
1−sin‌2θ=cos2θ, the integral simplifies to:
4. Back Substitution:
Since we substituted
x−1=sin‌θ, we have
θ=sin‌−1(x−1). Substituting back, we get:
θ+C=sin‌−1(x−1)+CTherefore, the correct answer is Option A:
sin‌−1(x−1)+C