What is ∫ (tan(x)sec^2(x)) dx equivalent to?

What is ∫ (tan(x)sec^2(x)) dx equivalent to?

• A. (1/2)tan^2(x) + C
• B. (1/2)tan(x) + C
• C. (tan(x)sec(x)) + C
• D. (sec^2(x)) + C

Answer: (A)

To determine the integral of the expression \( \int \tan(x) \sec^2(x) \, dx \), we can utilize the properties of derivatives and integrals from calculus. First, recall that the derivative of \( \tan(x) \) is \( \sec^2(x) \). This is critical because it allows us to recognize the form of the integrand. The product \( \tan(x) \sec^2(x) \) can be approached through substitution, using \( u = \tan(x) \). Consequently, the differential \( du = \sec^2(x) \, dx \). Now, we can rewrite the integral using this substitution: \[ \int \tan(x) \sec^2(x) \, dx = \int u \, du, \] where we substituted \( u \) for \( \tan(x) \) and \( du \) for \( \sec^2(x) \, dx \). The integral of \( u \) with respect to \( u \) is a straightforward application of the power rule: \[ \int u \, du = \frac{u^2}{2} + C = \frac{\tan^2(x)}

To determine the integral of the expression ( \int \tan(x) \sec^2(x) , dx ), we can utilize the properties of derivatives and integrals from calculus.

First, recall that the derivative of ( \tan(x) ) is ( \sec^2(x) ). This is critical because it allows us to recognize the form of the integrand. The product ( \tan(x) \sec^2(x) ) can be approached through substitution, using ( u = \tan(x) ). Consequently, the differential ( du = \sec^2(x) , dx ).

Now, we can rewrite the integral using this substitution:

[

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

]

where we substituted ( u ) for ( \tan(x) ) and ( du ) for ( \sec^2(x) , dx ).

The integral of ( u ) with respect to ( u ) is a straightforward application of the power rule:

[

\int u , du = \frac{u^2}{2} + C = \frac{\tan^2(x)}