What is the integral from 0 to pi/2 of x/(1+cos(x)) ?

The integral from 0 to pi/2 of x/(1+cos(x)) is pi/2-ln(2)

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integral from 0 to pi/2 of x/(1+cos(x))

\int_{0}^{\frac{π}{2}} \frac{x}{1 + cos ( x )} d x

\frac{π}{2} - ln (2)

Decimal Notation

0.87764 \dots

Solution steps

\int_{0}^{\frac{π}{2}} \frac{x}{1 + cos ( x )} d x

Apply Integration By Parts

: [x tan (\frac{x}{2}) - \int tan (\frac{x}{2}) d x]_{0}^{\frac{π}{2}}

= [x tan (\frac{x}{2}) - \int tan (\frac{x}{2}) d x]_{0}^{\frac{π}{2}}

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

= [x tan (\frac{x}{2}) - (- 2 ln cos (\frac{x}{2}))]_{0}^{\frac{π}{2}}

Simplify

= [x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})]_{0}^{\frac{π}{2}}

Compute the boundaries:

\frac{π}{2} - ln (2)

= \frac{π}{2} - ln (2)

\int_{0}^{3} \frac{1}{1 + x + 1} d x

\int_{0}^{2} 9 y^{2} y^{3} + 1 d y

\int_{1}^{2} \frac{1}{x} d x

\int_{0}^{3} (x - 3) e^{- x} d x

\int_{0}^{3} x 5^{x^{2}} d x

What is the integral from 0 to pi/2 of x/(1+cos(x)) ?
The integral from 0 to pi/2 of x/(1+cos(x)) is pi/2-ln(2)