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

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

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integral of t*sec^2(2t)

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

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

Solution steps

\int t sec^{2} (2 t) d t

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

y^{^{'}} - \frac{y}{t} = 0

\int \frac{cos ^{3} ( 4 x )}{sin ( 4 x )} d x

t^{3} + 2 t^{2} - t + 4

3 x^{2} y^{2} d x + (2 y^{3} + 4 y^{3}) d y = 0

\int_{0}^{3} e^{x^{2}} d x

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