Linear Regression

∫1/x dx = log x + c, ∫sec²x dx = tan x + c, ∫cos x dx = sin x + c

Step 1: ∫1/x dx = log x + c is correct. Step 2: ∫sec²x dx = tan x + c is correct. Step 3: ∫cos x dx = sin x + c is correct. Step 4: ∫x⁻¹ dx = log x + c, not x⁰ + c (power rule doesn't apply when n = -1).

½sin(x²) + c

Step 1: Let u = x², then du = 2x dx, so x dx = du/2. Step 2: ∫x cos(x²) dx = ∫cos u · (du/2) = ½∫cos u du. Step 3: = ½sin u + c. Step 4: Substitute back: ½sin(x²) + c.

½tan⁻¹(x/2) + c

Step 1: Use the formula ∫1/(x² + a²) dx = (1/a)tan⁻¹(x/a) + c. Step 2: Here x² + 4 = x² + 2², so a = 2. Step 3: ∫1/(x² + 4) dx = (1/2)tan⁻¹(x/2) + c = ½tan⁻¹(x/2) + c.

Integration by parts, Using LIATE rule

Step 1: This integral involves a product of algebraic function (x) and exponential function (e^x). Step 2: Integration by parts is the appropriate method. Step 3: LIATE rule helps choose which function to differentiate: L-I-A-T-E, so differentiate x (algebraic) and integrate e^x (exponential). Step 4: Substitution won't work as no function's derivative appears. Step 5: Direct integration is impossible for products. Step 6: Partial fractions apply to rational functions.