How to Solve the Integral of tan²(x) and Simplify it Using Trigonometric Identities

∫tan²(x)dx

To solve the integral ∫tan²(x)dx, we can use the trigonometric identity:

tan²(x) = sec²(x) – 1

So, the integral becomes:

∫(sec²(x) – 1)dx

Expanding this, we have:

∫sec²(x)dx – ∫dx

The first integral, ∫sec²(x)dx, can be easily solved by recognizing that it is the derivative of the tangent function:

∫sec²(x)dx = tan(x) + C

The second integral, ∫dx, represents the integral of a constant, which is simply the variable itself:

∫dx = x + C

Putting it all together, the final solution to the integral ∫tan²(x)dx is:

tan(x) – x + C

Where C is the constant of integration

To solve the integral ∫tan²(x)dx, we can use the trigonometric identity:

tan²(x) = sec²(x) – 1

So, the integral becomes:

∫(sec²(x) – 1)dx

Expanding this, we have:

∫sec²(x)dx – ∫dx

The first integral, ∫sec²(x)dx, can be easily solved by recognizing that it is the derivative of the tangent function:

∫sec²(x)dx = tan(x) + C

The second integral, ∫dx, represents the integral of a constant, which is simply the variable itself:

∫dx = x + C

Putting it all together, the final solution to the integral ∫tan²(x)dx is:

tan(x) – x + C

Where C is the constant of integration.

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