What is the integral of (e^x)/((cos(e^x))^2) ?

The integral of (e^x)/((cos(e^x))^2) is tan(e^x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (e^x)/((cos(e^x))^2)

\int \frac{e ^{x}}{( cos ( e ^{x} ) ) ^{2}} d x

tan (e^{x}) + C

Solution steps

\int \frac{e ^{x}}{( cos ( e ^{x} ) ) ^{2}} d x

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

Rewrite using trig identities

= \int \frac{e ^{x}}{cos ^{2} ( e ^{x} )} d x

Apply u-substitution

= \int sec^{2} (u) d u

Use the common integral:

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

= tan (u)

Substitute back

u = e^{x}

= tan (e^{x})

Add a constant to the solution

= tan (e^{x}) + C

What is the integral of (e^x)/((cos(e^x))^2) ?
The integral of (e^x)/((cos(e^x))^2) is tan(e^x)+C