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

User Jmibanez
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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]

User Madoka
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