Use u-substitution to determine the approximate value of integral 4, 0 e^x-e^-x/e^x+e^-x dx

Question

asked Oct 10, 2024 78.7k views

1 Answer

Madoka · Answer 1 · 2024-10-15T02:56:05+0000

The approximate value of the given integral is:

(1)/(2) [(e^4 + e^(-4))-2]

To evaluate the integral

$\int\limits^4_0 {(e^x - e^(-x))/(e^x + e^(-x)) } ,$

we can use the u-substitution method. Let
u = e^x + e^(-x) . then,
(du)/(dx) = e^x - e^(-x)

and solving for dx, we get
dx = (1)/(2) e^x + (1)/(2) e^(-1) du.

Now, we can substitute u and dx into the integral:

$\int\limits^4_0 {(e^x - e^(-x))/(e^x + e^(-x)) } dx= \int\limits^u_u (1)/(2) du$

To determine the new limits of integration, we substitute x=0 and x=4 into the expression for u:

$u(0)= e^0 +e^0=2\\u(4)= e^4 +e^(-4)$

Now, we can rewrite the integral in terms of u:

$\int\limits^(e^4+e^-4)_2 (1)/(2) du$

Integrating with respect to u, we get:

$\int\limits^(e^4+e^-4)_2 (1)/(2) du= (1)/(2) [u]^{e^4+e^(-4)}_2$

Substituting the limits:

(1)/(2) [(e^4 + e^(-4))-2]

So, the approximate value of the given integral is:

(1)/(2) [(e^4 + e^(-4))-2]