Q:

# How do you solve for the integral of sec(x)?

A:

Solve the integral of sec(x) by using the integration technique known as substitution. The technique is derived from the chain rule used in differentiation. The problem requires a knowledge of calculus and the trigonometric identities for differentiation.

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1. Multiply sec(x) by a value equal to one

In the integral, multiply sec(x) by (sec(x) + tan(x))/(sec(x) + tan(x)). Since the value is the same in both the numerator and denominator, it is equivalent to multiplying by one, which leaves the original value unchanged.

2. Set the value for u

Set the value of u as equal to sec(x) + tan(x).

3. Find the value of du

Based on the standard differential properties of trigonometric equations, du, the derivative of u, is set to (sec(x) * tan(x) + sec2(x)) dx.

4. Substitute in for u and du

Substitute u and du into the given equation to get du/u. Substitute into sec(x) by (sec(x) + tan(x))/(sec(x) + tan(x)) to get the equation (sec(x) * tan(x) + sec2(x))/(sec(x) * tan(x)).

5. Integrate du/u

Using basic integration rules, determine the integral of du/u as ln |u| + C.

6. Substitute out for u

In the equation ln |u| + C, replace the u with its given value of sec(x) + tan(x) to get the final solution of ln |sec(x) + tan(x)| + C.

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