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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