Derivatives and Integrals of Expressions with "e"

[Allanr13]

∫(e^(x)+e^(-x))^(2)

according to my paper, after this it should bring me to ∫(e^(2x)+2+e^(2x))
but i dont know which steps bring me there.

 

[Deleted member 4993]

∫(e^(x)+e^(-x))^(2)

according to my paper, after this it should bring me to ∫(e^(2x)+2+e^(2x))
but i dont know which steps bring me there.

Use:

(a + b)² = a² + b² + 2*a*b

 

[Allanr13]

ok so i see how it becomes (e^2x+e^2x)

but in respect of 2(a)(b)
does that mean 2(e^x)(e^-x)=2?

 

[Deleted member 4993]

ok so i see how it becomes (e^2x+e^2x)

but in respect of 2(a)(b)
does that mean 2(e^x)(e^-x)=2 yes


?

.

 

[Allanr13]

but how?

 

[HallsofIvy]

Hopefully you know that \(\displaystyle (a+ b)^2= a^2+ 2ab+ b^2\). For this problem you also need to know that, for any a (and, in particular, a= e), \(\displaystyle a^xa^y= a^{x+y}\).

So \(\displaystyle (e^x+ e^{-x})^2= (e^x)^2+ 2e^{x}e^{-x}+ (e^{-x})^2\).

Now, what are \(\displaystyle (e^x)^2\), \(\displaystyle e^xe^{-x}\), and \(\displaystyle (e^{-x})^2\)?

If you do not know those, then you seriously need to review algebra before continuing Calculus.