What is the integral of x/(1+cos(2x)) ?

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

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integral of x/(1+cos(2x))

\int \frac{x}{1 + cos ( 2 x )} d x

\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{x}{1 + cos ( 2 x )} d x

Apply Integration By Parts

: \frac{1}{2} x tan (x) - \int \frac{1}{2} tan (x) d x

= \frac{1}{2} x tan (x) - \int \frac{1}{2} tan (x) d x

\int \frac{1}{2} tan (x) d x = - \frac{1}{2} ln ∣ cos (x) ∣

= \frac{1}{2} x tan (x) - (- \frac{1}{2} ln ∣ cos (x) ∣)

Simplify

= \frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣

Add a constant to the solution

= \frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣ + C

arcsin (3 x)

\int \frac{x ^{2} - x + 6}{x ( x ^{2} + 3 )} d x

y^{^{''}} - (y^{^{'}})^{2} = 0

\frac{a x + b}{c x + d}

\frac{d}{d x} (3 x^{6})

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