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

The integral of 2/(cos(2x)) is ln|tan(2x)+sec(2x)|+C

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integral of 2/(cos(2x))

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

ln ∣ tan (2 x) + sec (2 x) ∣ + C

Solution steps

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

Rewrite using trig identities

= 2 \cdot \int sec (2 x) d x

Apply u-substitution

: \int sec (u) \frac{1}{2} d u

= 2 \cdot \int sec (u) \frac{1}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 2 \cdot \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 2 x

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

Simplify

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

= ln ∣ tan (2 x) + sec (2 x) ∣

Add a constant to the solution

= ln ∣ tan (2 x) + sec (2 x) ∣ + C

\frac{d}{d x} (\frac{x ^{2} + y - 1}{x + 1})

y \to 0 lim (4 y)

f (x) = x^{10}

x^{2} + 4 x - 7

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

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