x-sin-x-1-cos-x-dx-

Question Number 61056 by Tawa1 last updated on 28/May/19

\int \frac{x + \sin (x)}{1 + \cos (x)} dx

Answered by perlman last updated on 28/May/19

c o s (x) = 2 {c o s}^{2} (\frac{x}{2}) - 1

\frac{x + s i n (x)}{1 + c o s (x)} = \frac{x + s i n (x)}{2 {c o s}^{2} (\frac{x}{2})} = \frac{x}{2 {c o s}^{2} (\frac{x}{2})} + \frac{2 s i n (\frac{x}{2}) c o s (\frac{x}{2})}{2 {c o s}^{2} (\frac{x}{2})}

= \frac{x}{2} (1 + {t a n}^{2} (\frac{x}{2})) + \frac{s i n (\frac{x}{2})}{c o s (\frac{x}{2})}

==> \int \frac{x + s i n (x)}{1 + c o s (x)} d x = \int \frac{x}{2} (1 + {t g}^{2} (\frac{x}{2})) d x + \int \frac{s i n (\frac{x}{2})}{c o s (\frac{x}{2})} d x

\int \frac{s i n (\frac{x}{2})}{c o s (\frac{x}{2})} d x = - 2 l n ∣ c o s (\frac{x}{2}) ∣

\int \frac{x}{2} (1 + {t a n}^{2} (\frac{x}{2})) d x = x t a n (\frac{x}{2}) - \int t a n (\frac{x}{2}) d x = x t a n (\frac{x}{2}) + 2 l n ∣ c o s (\frac{x}{2}) ∣ + c

\int \frac{x + s i n (x)}{c o s (x) + 1} d x = x t a n (\frac{x}{2}) + c c c o n s t a n t \dots .

Commented by Tawa1 last updated on 28/May/19

God bless you sir

Answered by perlman last updated on 28/May/19

2 n d

x + s i n (x) = x ({c o s}^{2} (\frac{x}{2}) + {s i n}^{2} (\frac{x}{2})) + 2 s i n (\frac{x}{2}) c o s (\frac{x}{2})

1 + c o s (x) = 2 {c o s}^{2} (\frac{x}{2})

\int \frac{x + \sin (x)}{1 + \cos (x)} dx = \int \frac{x}{2} (1 + {t a n}^{2} (\frac{x}{2})) + t g (\frac{x}{2}) d x = \int d (x t a n (\frac{x}{2})) = x t a n (\frac{x}{2}) + c

Commented by Tawa1 last updated on 28/May/19

God bless you sir

Facebook Tweet Pin