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

The integral of (2e^xtan^2(e^x))/(sin(e^x)) is 2sec(e^x)+C

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integral of (2e^xtan^2(e^x))/(sin(e^x))

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

2 sec (e^{x}) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{e ^{x} tan ^{2} ( e ^{x} )}{sin ( e ^{x} )} d x

Apply u-substitution

: \int tan^{2} (u) csc (u) d u

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

Simplify

tan^{2} (u) csc (u) : sec (u) tan (u)

= 2 \cdot \int sec (u) tan (u) d u

Apply u-substitution

: \int 1 d v

= 2 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot v

Substitute back

= 2 \cdot 1 \cdot sec (e^{x})

Simplify

= 2 sec (e^{x})

Add a constant to the solution

= 2 sec (e^{x}) + C

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What is the integral of (2e^xtan^2(e^x))/(sin(e^x)) ?
The integral of (2e^xtan^2(e^x))/(sin(e^x)) is 2sec(e^x)+C