Understanding the Integration of sec²(x) using Trigonometric Identity and Power Rule

∫ sec²(x) dx

To integrate the function ∫ sec²(x) dx, we can use a trigonometric identity and the power rule of integration

To integrate the function ∫ sec²(x) dx, we can use a trigonometric identity and the power rule of integration.

Recall the identity: sec²(x) = 1 + tan²(x)

We can rewrite the integral as follows:
∫ sec²(x) dx = ∫ (1 + tan²(x)) dx

Now, we can split this integral into two separate integrals:
∫ (1 + tan²(x)) dx = ∫ 1 dx + ∫ tan²(x) dx

The integral of 1 dx is simply x, so this becomes:
∫ 1 dx = x

To integrate tan²(x), we use a trigonometric substitution. Let’s make the substitution u = tan(x), which means du = sec²(x) dx.

Replacing tan²(x) and dx with u² and du in the integral, we get:
∫ tan²(x) dx = ∫ u² du

Now, we can integrate u² using the power rule of integration:
∫ u² du = (1/3)u³ + C

Substituting back in u = tan(x), this becomes:
(1/3) tan³(x) + C

Finally, combining the results of the two integrals, we have:
∫ sec²(x) dx = x + (1/3) tan³(x) + C

So, the antiderivative of sec²(x) is x + (1/3) tan³(x) + C, where C is the constant of integration.

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