What is the indefinite integral sin(t)/t dt
look at this
www.youtube.com/watch?v=kXWyamlPCwY&ab_channel=GEORGEEULERBAITE
It's a special function, defined by this integral. Si(t)
To solve the indefinite integral of sin(t)/t dt, we will use a technique called integration by parts. This technique involves the product rule for differentiation in reverse.
Let's denote the integral as I:
I = ∫ (sin(t)/t) dt
Integration by parts states that the integral of the product of two functions u and v is equal to the product of one function u and the antiderivative of the other function v, minus the integral of the antiderivative of u times the derivative of v.
The formula for integration by parts is as follows:
∫ u * v dt = u * ∫ v dt - ∫ (u' * ∫ v dt) dt
In this case, we can choose:
u = sin(t) (u' = cos(t))
v = 1/t (v' = -1/t^2)
Now, let's apply the integration by parts formula:
I = ∫ (sin(t)/t) dt = u * ∫ v dt - ∫ (u' * ∫ v dt) dt
I = sin(t) * ∫ (1/t) dt - ∫ (cos(t) * ∫ (1/t) dt) dt
The first term, sin(t) * ∫ (1/t) dt, can be integrated directly:
∫ (1/t) dt = ln|t|
Replacing this back into the equation:
I = sin(t) * ln|t| - ∫ (cos(t) * ∫ (1/t) dt) dt
Now, we need to solve the remaining integral, ∫ (cos(t) * ∫ (1/t) dt) dt, using the same integration by parts technique:
Let's denote it as J:
J = ∫ (cos(t) * ∫ (1/t) dt) dt
Choose:
u = cos(t) (u' = -sin(t))
v = ∫ (1/t) dt (v' = 1/t)
Applying integration by parts:
J = cos(t) * ∫ (1/t) dt - ∫ (-sin(t) * ∫ (1/t) dt) dt
J = cos(t) * ln|t| + ∫ (sin(t) * ∫ (1/t) dt) dt
Comparing J with the original equation I, we can see that J is equivalent to -I:
J = -I
Substituting this back into the equation:
I = sin(t) * ln|t| - ∫ (cos(t) * ln|t|) dt
Now, we need to solve the integral ∫ (cos(t) * ln|t|) dt. This integral does not have a simple solution and cannot be expressed in terms of elementary functions. It is a special function known as the logarithmic integral, often denoted as Li(t).
Therefore, the indefinite integral ∫ (sin(t)/t) dt cannot be expressed using elementary functions and must be represented as:
I = sin(t) * ln|t| - Li(t) + C
where C is the constant of integration.