Trigonometric Identities Simplified | Evaluating the Integral of sec(x) dx

secx dx

The integral of sec(x) dx can be evaluated using the concept of trigonometric identities

The integral of sec(x) dx can be evaluated using the concept of trigonometric identities.

Recall the trigonometric identity: sec^2(x) = 1 + tan^2(x).

We can rewrite the integral as ∫sec(x) dx = ∫sec(x) * (1 + tan^2(x))/ (1 + tan^2(x)) dx.

Now, let’s consider the denominator 1 + tan^2(x). We can rewrite this denominator using the identity: 1 + tan^2(x) = sec^2(x). Therefore, the integral becomes:

∫sec(x) * (1 + tan^2(x))/ (sec^2(x)) dx

Simplifying further, we have:

= ∫(sec(x) * (1 + tan^2(x))/ (sec^2(x))) dx
= ∫(sec(x) * (1 + tan^2(x))/ (sec^2(x))) dx
= ∫(1 + tan^2(x)) dx

Breaking the integral into two parts, we have:

∫(dx) + ∫(tan^2(x) dx)

The integral of dx is simply x, so the first part simplifies to:

x

Now, let’s focus on the integral of tan^2(x) dx. This integral can be evaluated using the power rule for integration. Recall that the power rule states that the integral of x^n dx = (x^(n+1))/(n+1) + C.

Applying this rule, we have the integral:

∫(tan^2(x) dx) = ∫(tan^2(x) dx) = (tan^3(x))/(3) + C

Combining both parts, we get:

x + (tan^3(x))/(3) + C

Therefore, the integral of sec(x) dx is x + (tan^3(x))/(3) + C, where C is the constant of integration.

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