What is the integral of etan(x) ?

The integral of etan(x) is -eln|cos(x)|+C

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integral of etan(x)

\int e tan (x) d x

- e ln ∣ cos (x) ∣ + C

Solution steps

\int e tan (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= e \cdot \int tan (x) d x

Rewrite using trig identities

= e \cdot \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int - \frac{1}{u} d u

= e \cdot \int - \frac{1}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= e (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= e (- ln ∣ u ∣)

Substitute back

u = cos (x)

= e (- ln ∣ cos (x) ∣)

Simplify

= - e ln ∣ cos (x) ∣

Add a constant to the solution

= - e ln ∣ cos (x) ∣ + C

\frac{d ^{2}}{d x ^{2}} (cos^{2} (t))

\int \frac{sin ( 2 x )}{cos ^{2} ( 2 x ) + 5} d x

x \to 1 lim (4 x)

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

x \to 0 + lim (\frac{( e ^{x} )}{2 ^{x}})