C(T)=0.78125-0.5e^(-0.64T) (T+1.5625)

Using this equation, I don't understand 'Intergration by parts' to show this and what it means

Any help is appreciated!

Integration by parts is a method used in calculus to evaluate the integral of a product of two functions. It is based on the product rule for differentiation. The formula for integration by parts is:

∫ u dv = uv - ∫ v du

where u and v are functions, and du and dv are their differentials.

To understand integration by parts, let's consider the integral of a product of two functions, ∫ u dv. We want to rewrite this integral in a way that will be easier to evaluate. The basic idea is to split up the integral into two parts and choose u and dv appropriately.

In the given equation:

C(T) = 0.78125 - 0.5e^(-0.64T) (T+1.5625)

To compute the integral, we need to apply integration by parts. Let's consider the first term separately:

∫ 0.78125 d(T) = 0.78125T + C1

Here, we simply integrate the constant term (0.78125) with respect to T, which gives us (0.78125T). C1 represents the constant of integration.

Now, let's focus on the second term:

∫ -0.5e^(-0.64T) (T+1.5625) d(T)

We need to choose u and dv for applying integration by parts. In this case, we choose u as (T+1.5625) and dv as -0.5e^(-0.64T) d(T).

Taking the differentials of u and dv:

du = d(T)
v = ∫-0.5e^(-0.64T) d(T)

To find v, we need to integrate -0.5e^(-0.64T) with respect to T. This can be simplified by substituting u = -0.64T, so du = -0.64 d(T):

v = ∫ e^u (-0.64) du
v = -0.64 ∫ e^u du
v = -0.64 e^u

Now, we can apply the formula for integration by parts:

∫ -0.5e^(-0.64T) (T+1.5625) d(T) = (T+1.5625)(-0.64 e^(-0.64T)) - ∫ (-0.64 e^(-0.64T)) d(T)

Simplifying further:

= -(0.64(T+1.5625) e^(-0.64T)) - ∫ (0.64 e^(-0.64T)) d(T)

Now, we have another integral to evaluate, but notice that it is of the same form as the original integral. We can use integration by parts again to evaluate it, or if it is simpler or has a known solution, we can directly solve it.

By repeating the process of integration by parts, you can continue until you arrive at an integral that can be directly computed or until the integral becomes simpler or reaches a known form. The process of integration by parts allows you to break down complex integrals into simpler ones, making them easier to solve.