What is the integral of (2sec^2(x)-8sec^2(5x)) ?

The integral of (2sec^2(x)-8sec^2(5x)) is 2tan(x)-8/5 tan(5x)+C

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integral of (2sec^2(x)-8sec^2(5x))

\int (2 sec^{2} (x) - 8 sec^{2} (5 x)) d x

2 tan (x) - \frac{8}{5} tan (5 x) + C

Solution steps

\int 2 sec^{2} (x) - 8 sec^{2} (5 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 2 sec^{2} (x) d x - \int 8 sec^{2} (5 x) d x

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

\int 8 sec^{2} (5 x) d x = \frac{8}{5} tan (5 x)

= 2 tan (x) - \frac{8}{5} tan (5 x)

Add a constant to the solution

= 2 tan (x) - \frac{8}{5} tan (5 x) + C

What is the integral of (2sec^2(x)-8sec^2(5x)) ?
The integral of (2sec^2(x)-8sec^2(5x)) is 2tan(x)-8/5 tan(5x)+C