The integral of sec^2(x)

The integral of sec^2(x) is given by:

∫ sec^2(x) dx = tan(x) + C

where C is a constant.

This result can be derived using the following steps:

1. Recall that sec(x) = 1/cos(x), so sec^2(x) = 1/cos^2(x).
2. Use the identity cos^2(x) = 1 – sin^2(x) to rewrite sec^2(x) as follows:

sec^2(x) = 1/(1 – sin^2(x))

3. Use the substitution u = sin(x) to rewrite the integral as follows:

∫ sec^2(x) dx = ∫ 1/(1 – u^2) du

4. Integrate both sides:

∫ sec^2(x) dx = ∫ 1/(1 – u^2) du = tan^(-1)(u) + C = tan(x) + C

where C is a constant.

Here is some more information about the integral of sec^2(x):

• The integral of sec^2(x) is a common integral that arises in calculus and mathematical physics. It can be used to calculate the arc length of a curve, the surface area of a surface of revolution, and other quantities.
• The integral of sec^2(x) can be evaluated using the substitution u = sin(x). This substitution is based on the fact that sec^2(x) = 1/cos^2(x) and cos^2(x) = 1 – sin^2(x). By substituting u = sin(x), we can rewrite the integral in terms of u, which can then be easily integrated.
• The integral of sec^2(x) is an indefinite integral, which means that it is an antiderivative of sec^2(x) but it does not represent a specific function. To find the value of the integral for a specific value of x, we must also specify the value of the constant C.
• The integral of sec^2(x) can also be written in terms of the inverse tangent function, as shown in the previous step. The inverse tangent function, denoted as tan^(-1)(x), is the inverse function of the tangent function. It is defined as the function that returns the value of x for which tan(x) = y.