Evaluate integral e^3x cosh 2x dx.
A.1/2 e^5x + 1/2 e^x + C
B.1/10 e^5x + 1/2e^x + C<<< my answer
C.1/4e^3x + 1/2 x + C
D.1/10 e^5x + 1/5 x + C
Lets look at b.
d/dx (1/10 e^5x+ 1/2 e^x + C)
1/2 e^5x + 1/2 e^x
1/2 (e^3x (e^2x+e^-2x))
e^3x (e^2x+e-2x)/2
e^3x cosh2x
yep, you are correct.
To evaluate the integral ∫ e^3x cosh(2x) dx, you can use integration by parts.
Integration by parts is a technique that allows you to break down the integral of a product of functions into simpler integrals. The formula for integration by parts is:
∫ u dv = u v - ∫ v du
In this case, let's choose:
u = e^3x (function to differentiate)
dv = cosh(2x) dx (function to integrate)
To solve for du and v, we need to find the derivatives and integrals respectively:
du = d/dx(e^3x) dx = 3e^3x dx
To find v, we integrate dv = cosh(2x) dx. The integral of cosh(2x) is sinh(2x), so:
v = ∫ cosh(2x) dx = sinh(2x)
Now we can use the integration by parts formula:
∫ e^3x cosh(2x) dx = ∫ u dv = u v - ∫ v du
= e^3x sinh(2x) - ∫ sinh(2x) * 3e^3x dx
The new integral on the right side is simpler, as it involves only sinh(2x) and e^3x.
To evaluate this integral, you can perform another round of integration by parts or use trigonometric identities to simplify the expression.
Using integration by parts once again, let's set:
u = sinh(2x) (function to differentiate)
dv = 3e^3x dx (function to integrate)
Then we can find du and v:
du = d/dx(sinh(2x)) dx = 2cosh(2x) dx
v = ∫ 3e^3x dx = 3/3 * e^3x = e^3x
Applying the integration by parts formula again, we get:
∫ sinh(2x) * 3e^3x dx = sinh(2x) * e^3x - ∫ e^3x * 2cosh(2x) dx
The new integral on the right side can be further simplified. Using integration by parts once more, we set:
u = e^3x (function to differentiate)
dv = 2cosh(2x) dx (function to integrate)
Then compute du and v:
du = d/dx(e^3x) dx = 3e^3x dx
v = ∫ 2cosh(2x) dx = 2/2 * sinh(2x) = sinh(2x)
Applying the integration by parts formula one more time, we have:
∫ e^3x * 2cosh(2x) dx = e^3x * sinh(2x) - ∫ sinh(2x) * 3e^3x dx
Notice that this integral is the same as the one we started with, except for the sign. So we can substitute it back into the original equation:
∫ e^3x cosh(2x) dx = e^3x sinh(2x) - (e^3x sinh(2x) - ∫ sinh(2x) * 3e^3x dx)
This simplifies to:
∫ e^3x cosh(2x) dx = 2e^3x sinh(2x) + C
Therefore, the correct answer is option B: 1/10 e^5x + 1/2e^x + C.