Expand the expression inside the square root.

Here are the solutions for the integrals: 1. sqrt(1 + 2 x ( x + x))\,dx Step 1: Expand the expression inside the square root. 1 + 2 x ( x + x) = 1 + 2^2 x + 2 x x Step 2: Use the trigonometric identity 1 + ^2 x = ^2 x. 1 + 2^2 x + 2 x x = (1 + ^2 x) + ^2 x + 2 x x = ^2 x + ^2 x + 2 x x Step 3: Recognize the perfect square form (a+b)^2 = a^2 + b^2 + 2ab. Here, a = x and b = x. ^2 x + ^2 x + 2 x x = ( x + x)^2 Step 4: Substitute back into the integral and simplify the square root. sqrt(( x + x)^2)\,dx = | x + x|\,dx Assuming x + x 0 for the domain of integration, we can remove the absolute value. ( x + x)\,dx Step 5: Integrate each term. Recall that x\,dx = | x + x| + C and x\,dx = | x| + C. ( x + x)\,dx = | x + x| + | x| + C Step 6: Combine the logarithmic terms using A + B = (AB). | x ( x + x)| + C = |^2 x + x x| + C |^2 x + x x| + C 2. (x-1)/(x^2 - 2x + 26)\,dx Step 1: Use substitution. Let u = x^2 - 2x + 26. Step 2: Find the differential du. du = (d)/(dx)(x^2 - 2x + 26)\,dx = (2x - 2)\,dx = 2(x-1)\,dx From this, we have (x-1)\,dx = (1)/(2)\,du. Step 3: Substitute u and du into the integral. (1)/(u) · (1)/(2)\,du = (1)/(2) (1)/(u)\,du Step 4: Integrate with respect to u. (1)/(2) |u| + C Step 5: Substitute back u = x^2 - 2x + 26. The discriminant of x^2 - 2x + 26 is (-2)^2 - 4(1)(26) = 4 - 104 = -100, which is negative. Since the leading coefficient is positive, x^2 - 2x + 26 is always positive, so |x^2 - 2x + 26| = x^2 - 2x + 26. (1)/(2) (x^2 - 2x + 26) + C 3. (1)/(sqrt(4 - x^2))\,dx Step 1: Recognize this as a standard integral form for . The general form is (1)/(sqrt(a^2 - x^2))\,dx = ((x)/(a)) + C. Step 2: Identify a. In this integral, a^2 = 4, so a = 2. Step 3: Apply the formula. (1)/(sqrt(2^2 - x^2))\,dx = ((x)/(2)) + C ((x)/(2)) + C 3 done, 2 left today. You're making progress.