Solving the Integral of sec^2(x) using U-substitution | Step-by-Step Guide and Answer

integral of sec^2

The integral of sec^2(x) is a commonly asked question in calculus

The integral of sec^2(x) is a commonly asked question in calculus. To find the integral, we can use a technique called u-substitution.

First and foremost, let’s recall the identity for sec^2(x):
sec^2(x) = 1 + tan^2(x)

Now, we can begin with the integration process. Let’s start by introducing a new variable, u, and define it as u = tan(x). Therefore, du/dx = sec^2(x), which means du = sec^2(x) dx. We can rearrange this equation to solve for dx, yielding dx = du / sec^2(x).

Next, we substitute these new variables and the derived expression for dx back into the original integral:

∫ sec^2(x) dx = ∫ sec^2(x) (du / sec^2(x)).

Remarkably, sec^2(x) appears in both the numerator and denominator of the integrand, and they cancel each other out:

∫ sec^2(x) dx = ∫ du.

However, we need to adjust the limits of integration when applying the substitution. Since u = tan(x), when x = a (lower limit of integration), then u = tan(a), and when x = b (upper limit of integration), then u = tan(b).

Therefore, the new integral becomes:

∫ du = u + C,

where C is the constant of integration. Finally, we substitute tan(x) back in for u:

∫ sec^2(x) dx = tan(x) + C.

So, the integral of sec^2(x) is simply equal to tan(x) + C, where C represents the constant of integration.

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