What is the integral of (1-2e^{tan(x)})sec^2(x) ?

The integral of (1-2e^{tan(x)})sec^2(x) is tan(x)-2e^{tan(x)}+C

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integral of (1-2e^{tan(x)})sec^2(x)

\int (1 - 2 e^{t a n (x)}) sec^{2} (x) d x

tan (x) - 2 e^{t a n (x)} + C

Solution steps

\int (1 - 2 e^{t a n (x)}) sec^{2} (x) d x

Apply u-substitution

: \int 1 - 2 e^{u} d u

= \int 1 - 2 e^{u} d u

Apply the Sum Rule:

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

= \int 1 d u - \int 2 e^{u} d u

\int 1 d u = u

\int 2 e^{u} d u = 2 e^{u}

= u - 2 e^{u}

Substitute back

u = tan (x)

= tan (x) - 2 e^{t a n (x)}

Add a constant to the solution

= tan (x) - 2 e^{t a n (x)} + C

x \to 4 - lim (\frac{x + 2}{x - 4})

x \to 0 lim (\frac{lo g _{e} ( 1 + 3 x ^{2} )}{x ( e ^{x} - 1 )})

\frac{d}{d x} (sin^{2} (3 x - 2))

\frac{d}{d x} (4 arcsin (x))

\frac{d y}{d x} = 2 x (1 + y)

What is the integral of (1-2e^{tan(x)})sec^2(x) ?
The integral of (1-2e^{tan(x)})sec^2(x) is tan(x)-2e^{tan(x)}+C