What is the integral of x*sec^2(2x) ?

The integral of x*sec^2(2x) is 1/2 xtan(2x)+1/4 ln|cos(2x)|+C

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integral of x*sec^2(2x)

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

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

Solution steps

\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

= \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) ∣

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

Simplify

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

Add a constant to the solution

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

x^{2} y^{^{''}} + 3 x y^{^{'}} = 0

t \cdot \frac{d y}{d t} + 34 y = \frac{1}{t}

v + t v^{^{'}} = \frac{( 4 + v )}{( 4 v - 1 )}

y^{^{'}} = 7 x - y

f (x) = 3 x^{5} (x^{3} - 2 x)

What is the integral of x*sec^2(2x) ?
The integral of x*sec^2(2x) is 1/2 xtan(2x)+1/4 ln|cos(2x)|+C