Evaluate the integral of tan2(x)sec(x).

ELIAS, let's knock this out. Step 1: Use the trigonometric identity ^2 x = ^2 x - 1 to rewrite the integrand. ^2 x x \, dx = (^2 x - 1) x \, dx Step 2: Distribute x and split the integral into two parts. (^3 x - x) \, dx = ^3 x \, dx - x \, dx Step 3: Evaluate the integral of x. x \, dx = | x + x| Step 4: Evaluate the integral of ^3 x using integration by parts. Let I = ^3 x \, dx. We use integration by parts: u \, dv = uv - v \, du. Let u = x and dv = ^2 x \, dx. Then du = x x \, dx and v = x. I = x x - x ( x x) \, dx I = x x - x ^2 x \, dx Substitute ^2 x = ^2 x - 1 again: I = x x - x (^2 x - 1) \, dx I = x x - (^3 x - x) \, dx I = x x - ^3 x \, dx + x \, dx I = x x - I + | x + x| Now, solve for I: 2I = x x + | x + x| I = (1)/(2) x x + (1)/(2) | x + x| Step 5: Combine the results from Step 3 and Step 4. Substitute the expressions for ^3 x \, dx and x \, dx back into the equation from Step 2: ^3 x \, dx - x \, dx = ( (1)/(2) x x + (1)/(2) | x + x| ) - ( | x + x| ) + C Step 6: Simplify the expression. (1)/(2) x x + (1)/(2) | x + x| - | x + x| + C (1)/(2) x x - (1)/(2) | x + x| + C The final answer is (1)/(2) x x - (1)/(2) | x + x| + C. Send me the next one 📸