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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int \frac{1}{2} tan (2 x) d x = - \frac{1}{4} ln ∣ cos (2 x) ∣

= 2 (\frac{1}{2} x tan (2 x) - (- \frac{1}{4} ln ∣ cos (2 x) ∣))

Simplify

2 (\frac{1}{2} x tan (2 x) - (- \frac{1}{4} ln ∣ cos (2 x) ∣)) : x tan (2 x) + \frac{1}{2} ln ∣ cos (2 x) ∣

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

Add a constant to the solution

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

f (x) = - x^{2}

y^{^{'}} - 6 y = 5

\int \frac{5 x - 7}{( x ^{2} - x - 2 ) ( x - 3 )} d x

\frac{d}{d x} (6 ln (x) + 11)

\int_{- 0.5}^{1} cos (x^{2}) d x

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