evaluate the integral of
(e^3x)(cosh)(2x)(dx)
A.(1/2)(e^5x)+(1/2)(e^x)+C
B.(1/10)(e^5x)+(1/2)(e^x)+C
C.(1/4)(e^3x)+(1/2)(x)+C
D.(1/10)(e^5x)+(1/5)(x)+C
I did your last one by parts. Try the same trick again.
http://www.wolframalpha.com/widgets/gallery/view.jsp?id=a86aa57ade541fdb14f856fabd997a5e&sms_ss=googlebuzz&at_xt=4d11ee86a3a10396%252C0
Use for function
e^(3x)*cosh(2x)
use for variable
x
(e^3x)(cosh)(2x)(dx)
but cosh 2x = (1/2) (e^2x + e^-2x)
so
(1/2) [ e^5x + e^x ] dx
(1/10)e^5x + (1/2)e^x + c
so B
which we already knew from Wolfram
To evaluate the integral ∫ (e^3x)(cosh(2x)) dx, we can use integration by parts. The formula for integration by parts is ∫ u dv = uv - ∫ v du.
Let's choose u = (e^3x) and dv = cosh(2x) dx.
Taking the derivative of u, we get du/dx = 3(e^3x).
Integrating dv, we have v = ∫ cosh(2x) dx.
To find v, we can recognize that cosh(2x) is the derivative of sinh(2x) with respect to x. Therefore, v = sinh(2x)/2.
Using the formula for integration by parts, the integral becomes:
∫ (e^3x)(cosh(2x)) dx = u*v - ∫ v*du
= (e^3x)(sinh(2x)/2) - ∫ (sinh(2x)/2)*(3(e^3x)) dx
= (e^3x)(sinh(2x)/2) - (3/2) ∫ (e^3x)(sinh(2x)) dx.
Now, to evaluate the remaining integral on the right-hand side, we can use integration by parts again.
Let's choose again u = (e^3x) and dv = sinh(2x) dx.
Taking the derivative of u, we get du/dx = 3(e^3x).
Integrating dv, we have v = ∫ sinh(2x) dx.
To find v, we can recognize that sinh(2x) is the derivative of cosh(2x) with respect to x. Therefore, v = cosh(2x)/2.
Using the formula for integration by parts, the integral becomes:
∫ (e^3x)(sinh(2x)) dx = u*v - ∫ v*du
= (e^3x)(cosh(2x)/2) - ∫ (cosh(2x)/2)*(3(e^3x)) dx
= (e^3x)(cosh(2x)/2) - (3/2) ∫ (e^3x)(cosh(2x)) dx.
Now, we have a new integral that is the same as the original integral we started with.
Let's call this integral I. Therefore, we have:
I = (e^3x)(cosh(2x)/2) - (3/2)I.
To solve for I, we can rearrange the equation:
(5/2)I = (e^3x)(cosh(2x)/2).
Now, we can solve for I by dividing both sides of the equation by (5/2):
I = (2/5)(e^3x)(cosh(2x)/2).
Simplifying this expression, we get:
I = (1/5)(e^3x)(cosh(2x)).
Therefore, the value of the original integral is:
∫ (e^3x)(cosh(2x)) dx = (1/5)(e^3x)(cosh(2x)) + C, where C is the constant of integration.
Comparing this result with the given options, we see that the correct answer is D: (1/10)(e^5x) + (1/5)(x) + C.