What is the integral of e^xtan(e^x) ?

The integral of e^xtan(e^x) is -ln|cos(e^x)|+C

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integral of e^xtan(e^x)

\int e^{x} tan (e^{x}) d x

- ln ∣ cos (e^{x}) ∣ + C

Solution steps

\int e^{x} tan (e^{x}) d x

Apply u-substitution

: \int tan (u) d u

= \int tan (u) d u

Rewrite using trig identities

= \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

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

= \int - \frac{1}{v} d v

Take the constant out:

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

= - \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln ∣ cos (e^{x}) ∣

Add a constant to the solution

= - ln ∣ cos (e^{x}) ∣ + C

\frac{d}{d x} (a sin (2 x) + b cos (2 x))

\frac{d}{d x} (3 x^{2} + \frac{8 x}{3} - 2)

x \to 0 lim (- 5 (x - 3) ln (t (x) - 3))

\int ln (2 x) d x

\frac{\partial}{\partial x} (2 y \cdot e^{5 x^{2} - 9 y})

What is the integral of e^xtan(e^x) ?
The integral of e^xtan(e^x) is -ln|cos(e^x)|+C