How to Integrate the Function sec(x)tan(x) with Respect to x using U-Substitution

∫sec(x)tan(x)dx

To find the integral of sec(x)tan(x) with respect to x, we can use a technique called u-substitution

To find the integral of sec(x)tan(x) with respect to x, we can use a technique called u-substitution. Let’s start by identifying a part of the integrand to be our new variable, u.

Let u = sec(x)
Then, du = sec(x)tan(x)dx

We rewrite the integral using u:
∫sec(x)tan(x)dx = ∫du

Since du is a differential of u, the integral becomes:
∫du = u + C
Where C is the constant of integration.

Now we need to express u in terms of x to find the indefinite integral.

Using the identity: sec^2(x) – 1 = tan^2(x)
Rearranging this equation, we have: sec^2(x) = tan^2(x) + 1

Dividing both sides by sec(x) and taking the square root:
sec(x) = √(tan^2(x) + 1)

Substituting this value into our equation for u:
u = √(tan^2(x) + 1)

Therefore, our indefinite integral becomes:
∫sec(x)tan(x)dx = u + C = √(tan^2(x) + 1) + C

So, the integral of sec(x)tan(x) with respect to x is √(tan^2(x) + 1) + C.

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