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

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

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integral of x/(2cos^2(x))

\int \frac{x}{2 cos ^{2} ( x )} d x

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

Solution steps

\int \frac{x}{2 cos ^{2} ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x}{cos ^{2} ( x )} d x

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Simplify

\frac{1}{2} \cdot 2 (\frac{1}{2} x tan (x) - (- \frac{1}{2} ln ∣ cos (x) ∣)) : \frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣

= \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

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