## ∫sec u du

### To solve the integral ∫sec u du, we can use a technique called integration by substitution

To solve the integral ∫sec u du, we can use a technique called integration by substitution. Let’s go step by step through the solution:

Step 1: Identify the function to substitute. In this case, since the derivative of the secant function is the secant function multiplied by the tangent function, we can choose the substitution u = tan(x). This choice will help us simplify the integral.

Step 2: Calculate the derivative of u with respect to x. Since u = tan(x), du/dx = sec^2(x).

Step 3: Solve for dx. Rearranging the previous equation, we have dx = du / sec^2(x).

Step 4: Substitute u and dx in the integral. The integral becomes ∫sec(u) (du / sec^2(x)).

Step 5: Simplify. We can simplify sec^2(x) as 1 + tan^2(x), which can be rewritten using the substitution as sec^2(x) = 1 + u^2.

The integral becomes ∫sec(u) du / (1 + u^2).

Step 6: Use a new variable. Let’s use the variable v = u^2 + 1. This substitution will help us solve the integral.

Step 7: Calculate dv. Since v = u^2 + 1, differentiate both sides with respect to u:

dv/du = 2u.

Rewriting this equation, we have du = dv / (2u).

Step 8: Substitute v and du in the integral. The integral becomes ∫(sec(u)) (dv / (2u)).

Step 9: Simplify. Substituting the integral, we now have ∫ sec(u) dv / (2u).

Step 10: Divide by 2. We can divide the integral by 2 and rewrite it as (1/2) ∫ sec(u) dv / u.

Step 11: Let’s solve the integral. The integral ∫sec(u) dv/u is a well-known integral, equal to the natural logarithm of the absolute value of the quantity u + sec(u). Therefore, we have (1/2) ln |u + sec(u)| + C, where C is the constant of integration.

Step 12: Substitute back the variable u. Remember that u = tan(x), so we have (1/2) ln |tan(x) + sec(tan(x))| + C, where C is the constant of integration.

And there you have it! The integral of sec u du is (1/2) ln |tan(x) + sec(tan(x))| + C.

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