Calculate ∫ (tan(x))^2 dx.

Calculate ∫ (tan(x))^2 dx.

• A. tan(x) - x + C
• B. 2tan(x) + C
• C. ln|sec(x)| + C
• D. sec(x) + C

Answer: (A)

To calculate the integral of \((\tan(x))^2\), we can use a fundamental identity from trigonometry: \[ \tan^2(x) = \sec^2(x) - 1. \] By substituting this identity into the integral, we have: \[ \int \tan^2(x) \, dx = \int (\sec^2(x) - 1) \, dx. \] This can be split into two separate integrals: \[ \int \tan^2(x) \, dx = \int \sec^2(x) \, dx - \int 1 \, dx. \] The integral of \(\sec^2(x)\) is a well-known result: \[ \int \sec^2(x) \, dx = \tan(x). \] On the other hand, the integral of \(1\) with respect to \(x\) is simply: \[ \int 1 \, dx = x. \] Combining these results gives: \[ \int \tan^2(x) \, dx = \tan(x) - x + C, \] where \(C\) is the constant

To calculate the integral of ((\tan(x))^2), we can use a fundamental identity from trigonometry:

[

\tan^2(x) = \sec^2(x) - 1.

]

By substituting this identity into the integral, we have:

[

\int \tan^2(x) , dx = \int (\sec^2(x) - 1) , dx.

]

This can be split into two separate integrals:

[

\int \tan^2(x) , dx = \int \sec^2(x) , dx - \int 1 , dx.

]

The integral of (\sec^2(x)) is a well-known result:

[

\int \sec^2(x) , dx = \tan(x).

]

On the other hand, the integral of (1) with respect to (x) is simply:

[

\int 1 , dx = x.

]

Combining these results gives:

[

\int \tan^2(x) , dx = \tan(x) - x + C,

]

where (C) is the constant