Answer to Question #332665 in Calculus for Olga

Question

Determine the following integral

∫  (1+cos^2 x  / 1+cos 2x ) dx

a.    1 / 2 [ tan x + x] + c

b.   1 / 2[1− 1 / cos x ] + c

c.    1 / 2 + c

d.   1 / 2 tan x + x + c

Answer 1

$$\int \dfrac{1+\cos^2x}{1+\cos2x}dx=\int \dfrac{1+\cos^2x}{2\cos^2x}dx$$

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

$$=\dfrac{1}{2}(\tan x+x)+C$$

a. $\dfrac{1}{2}(\tan x+x)+C$