Calculus Examples

$e^{2 x} + 2 + e^{- 2 x}$

Step 1

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

${(e^{x})}^{2} + 2 + e^{- 2 x}$

Step 2

Rewrite $e^{- 2 x}$ as ${(e^{- x})}^{2}$ .

${(e^{x})}^{2} + 2 + {(e^{- x})}^{2}$

Step 3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 = 2 \cdot e^{x} \cdot e^{- x}$

Step 4

Rewrite the polynomial.

${(e^{x})}^{2} + 2 \cdot e^{x} \cdot e^{- x} + {(e^{- x})}^{2}$

Step 5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = e^{x}$ and $b = e^{- x}$ .

${(e^{x} + e^{- x})}^{2}$