What is the integral of 4xsec^2(2x) ?

The integral of 4xsec^2(2x) is 2xtan(2x)+ln|cos(2x)|+C

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integral of 4xsec^2(2x)

\int 4 x sec^{2} (2 x) d x

2 x tan (2 x) + ln ∣ cos (2 x) ∣ + C

Solution steps

\int 4 x sec^{2} (2 x) d x

Take the constant out:

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

= 4 \cdot \int x sec^{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

= 4 (\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) ∣

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

Simplify

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

= 2 x tan (2 x) + ln ∣ cos (2 x) ∣

Add a constant to the solution

= 2 x tan (2 x) + ln ∣ cos (2 x) ∣ + C

\frac{- 5 e ^{(- s)}}{( s ^{2} + 6 s + 25 )}

f (x) = ln (sech (2 x))

\int x^{2} sin (3 x) d x

\frac{d y}{d x} = tan^{2} (x)

What is the integral of 4xsec^2(2x) ?
The integral of 4xsec^2(2x) is 2xtan(2x)+ln|cos(2x)|+C