Evaluate!!!!
integral -10 to 10 ((2e^x)/(sinhx+coshx))dx
Thanks!
I need someone to show me steps. I used wolframalpha and got 40 but don't know how to solve this
Thanks!!!
To evaluate the integral ∫(2e^x)/(sinhx+coshx) dx from -10 to 10, we can use the "odd/even" property of hyperbolic trigonometric functions.
First, let's simplify the integrand by rewriting sinh(x) and cosh(x) in terms of exponential functions:
sinh(x) = (e^x - e^(-x))/2
cosh(x) = (e^x + e^(-x))/2
Substituting these expressions back into the integrand, we get:
(2e^x)/((e^x - e^(-x))/2 + (e^x + e^(-x))/2)
Now simplify this expression:
(2e^x)/((e^x - e^(-x) + e^x + e^(-x))/2)
(2e^x)/(2e^x/2)
2e^x/e^x
Canceling out the exponentials, we are left with:
2
Now we can evaluate the integral by integrating the constant term:
∫(2)dx = 2x
To find the value of the definite integral from -10 to 10, we subtract the value of the antiderivative at the lower bound from the value at the upper bound:
[2x] from -10 to 10
(2 * 10) - (2 * -10)
20 + 20
40
Therefore, the value of the given integral is 40.
To evaluate the integral ∫-10 to 10 ((2e^x)/(sinhx+coshx)) dx, you can follow these steps:
Step 1: Simplify the integral
We notice that the numerator 2e^x and the denominator sinhx+coshx both have the common factor of e^x. We can simplify the integral as follows:
∫-10 to 10 ((2e^x)/(sinhx+coshx)) dx = ∫-10 to 10 (2/(sinhx/coshx+1)) dx
Step 2: Use the hyperbolic trigonometric identity
The hyperbolic trigonometric identity relates sinh and cosh:
sinh^2x + cosh^2x = 1
By rearranging this equation, we can express coshx in terms of sinhx:
cosh^2x = 1 + sinh^2x
coshx = sqrt(1 + sinh^2x)
Substituting this expression into our integral:
∫-10 to 10 (2/(sinhx/(sqrt(1 + sinh^2x)) + 1)) dx
Step 3: Make a substitution
Let u = sinhx. Then du = coshx dx. Also, x = sinh^(-1)(u) and dx = du / coshx.
Substituting these into the integral:
∫ ((2/((u/(sqrt(1 + u^2)) + 1))) * (du / coshx))
Simplifying further:
∫ (2/((u + sqrt(1 + u^2))/sqrt(1 + u^2))) du
= 2 ∫ (sqrt(1 + u^2) / (u + sqrt(1 + u^2))) du
Step 4: Use a trigonometric substitution
Let u = sinh(t), then du = cosh(t) dt.
Substituting these into the integral:
2 ∫ (sqrt(1 + (sinh(t))^2) / (sinh(t) + sqrt(1 + (sinh(t))^2))) cosh(t) dt
= 2 ∫ (sqrt(cosh^2(t)) / (sinh(t) + sqrt(cosh^2(t)))) cosh(t) dt
= 2 ∫ (cosh(t) / (sinh(t) + cosh(t))) dt
Step 5: Simplify the integrand
Using the hyperbolic trigonometric identity cosh(t) = sqrt(1 + sinh^2(t)), we can simplify the integrand further:
2 ∫ (sqrt(1 + sinh^2(t)) / (sinh(t) + sqrt(1 + sinh^2(t)))) dt
= 2 ∫ (sqrt(1 + sinh^2(t)) / (sinh(t) + sqrt(1 + sinh^2(t)))) dt
Step 6: Evaluate the integral
The resulting integral is a function of t, so you can evaluate it using numerical methods or a symbolic solver.
Please note that the actual evaluations of the integral may involve more advanced mathematics and may require numerical approximation methods depending on the specific input function.
recall that
sinhx = (e^x-e^-x)/2
coshx = (e^x+e^-x)/2
so,
sinhx+coshx = e^x
2e^x/e^x = 2