How to Integrate Secx | Step-by-Step Guide to Finding the Integral of Secant Function

int secx

The term “int secx” seems to be an incomplete expression since “int” may refer to “integral” and “secx” may refer to the secant function

The term “int secx” seems to be an incomplete expression since “int” may refer to “integral” and “secx” may refer to the secant function. Assuming you are asking about integrating the secant function (∫secx dx), let’s go through the process step by step.

To integrate secx, we can use a technique called integration by parts or by using a substitution. Let’s use the substitution method.

1. Let’s start by choosing a substitution: u = secx + tanx. This implies that du = (secx * tanx + sec^2x) dx.

2. Rearrange the equation to express dx in terms of du: dx = du / (secx * tanx + sec^2x).

3. Substitute the values of u and dx back into the integral:
∫secx dx = ∫(1/(secx * tanx)) * (du / (secx * tanx + sec^2x)).

4. Simplify the integral using the substitution:
∫secx dx = ∫(du / (u * tanx + 1)).

5. Next, we can use a trigonometric identity to further simplify this expression. Note that tanx + 1 = secx, so the integral becomes:
∫(du / u).

6. Now, we can integrate the simplified expression:
∫(du / u) = ln|u| + C.

7. Substitute the value of u back in terms of x:
∫secx dx = ln|secx + tanx| + C.

So, the integral of secx with respect to x is ln|secx + tanx| + C, where C is the constant of integration.

Note: Sometimes, this result is written as ln|secx + tanx| + ln|secx| + C, where ln|secx| can also be expressed as ln|secx + tanx – tanx|. Both expressions are equivalent.

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