Solving the Integral of Tan²(x): Step-by-Step Guide with Trigonometric Identities and Integration Techniques

∫tan²(x)dx

To solve the integral of tan²(x), we need to use trigonometric identities and integration techniques

To solve the integral of tan²(x), we need to use trigonometric identities and integration techniques.

First, we can express tan²(x) in terms of sine and cosine using the identity: tan²(x) = sec²(x) – 1.

So, we have:
∫tan²(x)dx = ∫(sec²(x) – 1)dx

Now, let’s break it down into two integrals:

∫sec²(x)dx – ∫1dx

The integral of sec²(x) can be found using the basic integral formula: ∫sec²(x)dx = tan(x) + C, where C is the constant of integration.

So, the first integral becomes: ∫sec²(x)dx = tan(x)

For the second integral, ∫1dx is simply x + C.

Now, let’s put everything together:

∫tan²(x)dx = ∫(sec²(x) – 1)dx
= tan(x) – (x + C)

So, the final answer is: ∫tan²(x)dx = tan(x) – x + C, where C is the constant of integration.

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