Calculus 2 evaluate the integral (7.4 7)

Question

Evaluate the integral. (Use C for the constant of integration.)

2dx/2 + ex (numerator/denominator). Please show detailed work an box the final answer, thank you

Answer 1

let x = ln U ---> exp(x) = U

then dx = du/U

the integral becomes: du/U* 1/(2+ U) with the 2 factored out of the integral

Partial Fraction Decomposition:

1/ [ U(U+2) ] = a/U + b/(U+2)

multiplies both sides by U(U+2):

1 = a(U+2) + bU

= aU + 2a + bU

= (a+b)U + 2a

(a+b) = 0 and 2a = 1

Then a=1/2, so 1/2 +b = 0 ---> b = -1/2

THe partial fraction decomposition is : (1/2) (1/U) + (-1/2)(1/(U+2))

integrating....

(1/2) ln U - (1/2) ln (U+2) =

(1/2) ln | exp(x)| - (1/2) ln | exp(x) + 2 |

Note that the 2 kills the 1/2

x - ln | exp(x)+2 |+ C

check by differentiation:

1 - exp(x)/[exp(x)+2] =

[exp(x)+2 - exp(2)] / [exp(x)+2 ] =

2 / [exp(x)+2 ] <-- yes, it checks

The general anti-derivative is x - ln | exp(x)+2 |+ C