Question

integrate by substitution:
(e^x + 1/e^x) ^2
please show all the steps..

Answer 1

You really have to do it by substitution? because I had an easier set of steps.

1) re-write the 1/e^x as e^(-x).
2) use (a+b)²=a²+2ab+b²
3) Integrate the sum term by term
optional 4) After you get every you can factor out of 1/2.

Answer 2

after you get every *integral

Answer 3

actually its the ques to do it by substitution.. please help me through substitution.. @SolomonZelman

Answer 4

\[\int\limits_{ }^{ } \left(\begin{matrix} e^x+\frac{1}{e^x} \end{matrix}\right)^2~dx\]\[\int\limits_{ }^{ } \left(\begin{matrix} e^x+e^{-x} \end{matrix}\right)^2 ~dx\]\[\int\limits_{ }^{ } \left(\begin{matrix} e^{-x}+e^x \end{matrix}\right)^2 ~dx\]\[\int\limits_{ }^{ } \left(\begin{matrix} e^{-2x}+e^{2x}+2 \end{matrix}\right) ~dx\]\[\int\limits_{ }^{ } e^{-2x}~dx+\int\limits_{ }^{ } e^{2x}~dx+\int\limits_{ }^{ } 2~dx \]

Answer 5

I am not very sure what and where to sub in this case... and esp, why

Answer 6

ohk. thanks anyway.. btw i knew this method.. i did it by this but the ques states do it by substitution.. @ganeshie8 can u please help me??

Answer 7

maybe sub in u for e^x ?

Answer 8

The you would be getting the same but with a 'u'\[\int\limits_{ }^{ } u^{-2}~du+\int\limits_{ }^{ } u^{2}~du+\int\limits_{ }^{ } 2~du\]

Answer 9

Then*

Answer 10

yeah there no difference.. if i substitute 1/e^x then?

Answer 11

basically, no

Answer 12

then?

Answer 13

I mean "then" after you substitute `u` for e^x, you would be getting the same thing, except that you are wasting a step by doing that.

Answer 14

and what if i substitute 1/e^x??

Answer 15

Well, basically i it will be like this, (using § for an integral sign)

§ (e^x + 1/e^x)² dx
§ (u + 1/u)² du
§ (u + u\(\scriptsize\color{slate}{^{-1}}\) )² du
§ (u² + u\(\scriptsize\color{slate}{^{-2}}\)+2 ) du
§ u² du + § u\(\scriptsize\color{slate}{^{-2}}\) du + § 2 du

Answer 16

the rest you can do yourself, and sub e^x for you when you are done...
let me know if you need more help

Answer 17

ohk.. thanks (:

Answer 18

yw