Question

Evaluate the following integral.
∫d x/x²-2 x+37

Answer 1

To evaluate the integral ∫dx/(x²-2x+37), complete the square in the denominator and make a substitution. The final answer is (1/6)arctan((x-1)/6) + C.

Step-by-step explanation:

To evaluate the integral ∫dx/(x²-2x+37), we can use the method of completing the square. First, we complete the square in the denominator: x²-2x+37 = (x-1)²+36. So, the integral becomes ∫dx/((x-1)²+36). To simplify this, we can make a substitution by letting u = x-1. Then, dx = du and the integral becomes ∫du/((u)²+36). This integral can be evaluated using inverse tangent function: ∫du/((u)²+36) = (1/6)arctan(u/6) + C Substituting back, the final answer is (1/6)arctan((x-1)/6) + C.