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

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)}