Simplifying the Integration of ∫sec(x)dx using Integration by Substitution

∫sec(x)dx

To integrate ∫sec(x)dx, we can use a technique called integration by substitution

To integrate ∫sec(x)dx, we can use a technique called integration by substitution.

Let’s begin by selecting a substitution for the integral. In this case, a good choice is u = sec(x) + tan(x). We can differentiate u with respect to x to find du/dx.

du/dx = d/dx(sec(x) + tan(x))
= sec(x)tan(x) + sec^2(x)
= sec(x)(tan(x) + sec(x))
= sec(x)u

Now, let’s rearrange the equation to solve for dx:

dx = du/(sec(x)u)

Next, we substitute our new variables and equation into the integral:

∫sec(x)dx = ∫(1/sec(x))dx = ∫(1/u)dx

Notice that the original integral can be simplified to the integral of 1/u with respect to x.

Now, we can rewrite the integral using our substitutions:

∫(1/u)dx = ∫(1/u) * (du/(sec(x)u))

We can cancel out the u’s and simplify:

∫(1/u) * (du/(sec(x)u)) = ∫(1/(sec(x)u))du
= ∫(1/(u^2))du

The integral of 1/u^2 with respect to u is -1/u. Therefore:

∫sec(x)dx = -1/u + C

Now, we need to substitute u back in terms of x:

-1/u = -1/(sec(x) + tan(x))

Therefore, the final answer to ∫sec(x)dx is:

∫sec(x)dx = -1/(sec(x) + tan(x)) + C

Note: C represents the constant of integration, which can take any value.

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