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

The integral of 4xsec^2(x)-4xtan^2(x) is 2x^2+C

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integral of 4xsec^2(x)-4xtan^2(x)

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

2 x^{2} + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 4 x sec^{2} (x) d x = 4 (x tan (x) + ln ∣ cos (x) ∣)

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

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

Simplify

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

= 2 x^{2}

Add a constant to the solution

= 2 x^{2} + C

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