Step-by-Step Guide: Finding the Integral of sec(x)tan(x) dx Using Substitution Method

∫secxtanxdx

To find the integral of sec(x)tan(x) dx, we can use a substitution method

To find the integral of sec(x)tan(x) dx, we can use a substitution method. Let’s try substituting u = sec(x).

First, find the derivative of u with respect to x:
du/dx = sec(x)tan(x) dx

We have sec(x)tan(x) dx in the original integral and we have du/dx in terms of sec(x)tan(x) dx. Thus, we can rewrite the integral as:

∫sec(x)tan(x) dx = ∫du/dx dx

Now, we can rewrite the integral using substitution:

∫du/dx dx = ∫du

Since du/dx = sec(x)tan(x) dx, we can simplify the integral to:

∫sec(x)tan(x) dx = ∫du

Integrating du is straightforward:

∫du = u + C

Recall that u = sec(x), so substitute that back in:

∫sec(x)tan(x) dx = sec(x) + C

Thus, the integral of sec(x)tan(x) dx is sec(x) + C, where C is the constant of integration.

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