Question

Complete the square in the denominator, make an appropriate substitution, and integrate. (Hint: 2x−5).

a. ∫1/ (2x−5)² dx.
b. ∫ 1/ 2(x−5/2) dx.
c. ∫ 1/ 4x−10 dx.
d. ∫ 1/2xdx.

Answer 1

To integrate the expression ∫1/ (2x−5)² dx, complete the square in the denominator, make the substitution u = x-5/2, and integrate the expression (1/4)(1/u^2) du.

Step-by-step explanation:

To integrate the expression ∫1/ (2x−5)² dx, we can complete the square in the denominator and make an appropriate substitution. Let's start by completing the square in (2x-5)²:

(2x-5)² = 4(x-5/2)²

Now we can make the substitution u = x-5/2, which gives du = dx.

Substituting this into the integral, we get ∫1/4u² du. Integrating this expression results in (1/4) * (1/u) + C.

Finally, substituting back u = x-5/2 gives the final answer: (1/4) * (1/(x-5/2)) + C.