Integral of 1+sin(x) all over cos(x)^2

[Giuseppe]

I was wondering what the best way to do this integral was:

Integral of 1+sin(x) all over cos(x)^2

Is subsitution the best way?

 

[hypermorphism]

The easiest way is to write the fraction as the sum of fractions ((a+b)/c = a/c + b/c)) and integrate the first and second terms of the sum separately. One is a basic antiderivative and the other is a simple substitution.

 

[TD]

So what hypermorphism means is that you do:

$$\int {\frac{{1 + \sin x}}{{\cos ^2 x}}} dx = \int {\frac{1}{{\cos ^2 x}}} dx + \int {\frac{{\sin x}}{{\cos ^2 x}}} dx$$

As he said, the first one is a standard antiderivative, for the second one try $y = \cos x$