definite integral of (e^(8x))cosh(8x)dx from 0 to 1/16
-I substitued the definition of cosh in, i.e. (e^(8x) + e^(-8x)/2 but i cant get the "e's" to combine and i know the answer is e/32
you had
∫ e^(8x) (e^(8x) + e^(-8x) )/2 dx
which is
= ∫ ( e^(16x) + e^0 )/2 dx
= (1/2)(1/16)e^(16x) + (1/2)x | from 0 to 1/16
= (1/2)(1/16) e^1 + 1/32 - (1/2)(1/16)e^0 - 0
= (1/32)e
or
= e/32
To solve this definite integral, we can simplify the integrand by combining the exponential terms before integrating.
The integrand is (e^(8x))cosh(8x)dx.
To combine the exponential terms, we can use the identity cosh(x) = (e^x + e^(-x))/2.
So, we substitute the definition of cosh into the integrand:
(e^(8x)) * (e^8x + e^(-8x))/2 dx
The constant factor of 1/2 can be brought outside the integral:
1/2 * (e^(8x)) * (e^8x + e^(-8x)) dx
Now, let's simplify the expression:
1/2 * (e^(8x + 8x) + e^(8x - 8x)) dx
Simplifying the exponents:
1/2 * (e^(16x) + e^(0)) dx
Since e^0 is equal to 1, we can simplify further:
1/2 * (e^(16x) + 1) dx
Now we can integrate:
∫[0, 1/16] (1/2 * (e^(16x) + 1)) dx
Let's evaluate the integral:
(1/2) * ∫[0, 1/16] (e^(16x) + 1) dx
To find the antiderivative of e^(16x), we divide by the coefficient of x, that is 16:
(1/2) * (1/16) * e^(16x) + (1/2) * x + C
Now, we can evaluate the integral with the limits of integration:
(1/2) * (1/16) * e^(16 * (1/16)) + (1/2) * (1/16) - (1/2) * (1/16) * e^(0) - (1/2) * 0
Simplifying further:
(1/2) * (1/16) * e + (1/32) - (1/32) * 1
(1/32) * e + 0
The definite integral of (e^(8x))cosh(8x)dx from 0 to 1/16 is equal to (1/32) * e, which matches the expected answer of e/32.