What is the integral of x/(1+cos(x)) ?

The integral of x/(1+cos(x)) is xtan(x/2)+2ln|cos(x/2)|+C

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integral of x/(1+cos(x))

integral

\frac{x}{1 + cos ( x )}

x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2}) + C

Solution steps

\int \frac{x}{1 + cos ( x )} d x

Apply Integration By Parts

: x tan (\frac{x}{2}) - \int tan (\frac{x}{2}) d x

= x tan (\frac{x}{2}) - \int tan (\frac{x}{2}) d x

\int tan (\frac{x}{2}) d x = - 2 ln cos (\frac{x}{2})

= x tan (\frac{x}{2}) - (- 2 ln cos (\frac{x}{2}))

Simplify

= x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})

Add a constant to the solution

= x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2}) + C

1 + cos^{2} (x)

f (x) = x^{9}

f (x) = x^{3}

\frac{1}{( x ^{2} + 2 x + 5 )}

(\frac{x}{5})^{2}

What is the integral of x/(1+cos(x)) ?
The integral of x/(1+cos(x)) is xtan(x/2)+2ln|cos(x/2)|+C