## ∫ sec(x) tan (x) dx

### To solve the integral ∫ sec(x) tan(x) dx, we can use a technique called integration by substitution

To solve the integral ∫ sec(x) tan(x) dx, we can use a technique called integration by substitution.

Let’s start by letting u = sec(x), implying that du = sec(x) tan(x) dx. Rearranging this equation gives us dx = du/(sec(x) tan(x)).

Now we substitute these values in the integral:

∫ sec(x) tan(x) dx = ∫ (1/u) du/(sec(x) tan(x)).

Next, we observe that the denominator, sec(x) tan(x), can be simplified using trigonometric identities. We know that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x). So, sec(x) tan(x) = (1/cos(x)) * (sin(x)/cos(x)). Simplifying further, we get sec(x) tan(x) = sin(x)/cos^2(x).

Substituting this in our integral, we have:

∫ (1/u) du/(sin(x)/cos^2(x)).

Now, we can simplify this as:

∫ (du/u) * (cos^2(x)/sin(x)).

Next, we observe that cos^2(x) is the same as 1 – sin^2(x) by the Pythagorean identity. Thus, we can substitute this value in the integral:

∫ (du/u) * ((1 – sin^2(x))/sin(x)).

Now, we can split this integral into two separate integrals:

∫ (du/u) – ∫ (sin^2(x) du/u sin(x)).

Simplifying further, we have:

∫ (du/u) – ∫ (sin(x) du/u).

The first integral, ∫ (du/u), is simply ln|u| + C, where C is the constant of integration.

The second integral, ∫ (sin(x) du/u), can be solved by substituting du/u as -d(cos(x)). Rearranging, this gives us ∫ -sin(x) d(cos(x)) = -∫ d(cos(x)) = -cos(x) + D, where D is another constant of integration.

Combining both integrals, we get:

∫ sec(x) tan(x) dx = ln|u| – cos(x) + D,

where u = sec(x).

Finally, we can substitute back u = sec(x):

∫ sec(x) tan(x) dx = ln|sec(x)| – cos(x) + D.

And there you have it! The value for the integral ∫ sec(x) tan(x) dx is ln|sec(x)| – cos(x) + D.

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