Simplifying the integral: solving ∫ sec^2 dx using trigonometric identities

sec^2 dx

The expression “sec^2 dx” represents the integral of the square of the secant function with respect to x

The expression “sec^2 dx” represents the integral of the square of the secant function with respect to x.

To solve this integral, we will use a trigonometric identity. The identity for the derivative of the secant function is:

d(sec x) / dx = sec x * tan x

We can rewrite this identity as:

sec x * tan x = d(sec x)

Now, let’s rewrite the integral expression in terms of this identity:

∫ sec^2 dx = ∫ sec x * sec x * dx

Using the identity we derived, we can rewrite the integral as:

∫ sec x * sec x * dx = ∫ sec x * tan x * sec x * dx

Now, we have a product of two secant functions and one tangent function. We can simplify this further using another trigonometric identity. The identity for the derivative of the tangent function is:

d(tan x) / dx = sec^2 x

We can rewrite this identity as:

sec^2 x = d(tan x)

Now, let’s rewrite the integral expression again using this identity:

∫ sec x * tan x * sec x * dx = ∫ sec x * d(tan x)

We can now integrate both sides of the equation:

∫ sec x * d(tan x) = tan x + C

Where C is the constant of integration.

Therefore, the solution to the integral ∫ sec^2 dx is tan x + C.

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