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First Fundamental Theorem of Calculus

Integrate ∫...

Question

Integrate $\int \frac{sin x + 1}{1 - sin x} d x$

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Solution

$\int \frac{1 + s i n x}{1 - s i n x} d x$
$= \int \frac{(1 + s i n x) (1 + s i n x)}{(1 - s i n x) (1 + s i n x)} d x$
$= \int \frac{(1 + s i n x)^{2}}{1 - s i n^{2} x} d x$
$= \int \frac{s i n^{2} x + 1 + 2 s i n x}{c o s^{2} x} d x$
$= \int (t a n^{2} x + s e c^{2} x + 2 s e c x t a n x) d x$
$= \int (2 s e c^{2} x - 1 + 2 s e c x t a n x) d x$
$= 2 t a n x - x + s e c x + C$

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