can sombody explain how (0to q)∫[(q-CE)/rc]dt become (0to q)∫[1/(q-CE)]dt= (0to t)∫[1/rc]dt and not
(0to q)∫(q-CE)dt= (0to t)∫[1/rc]dt. I don't understand where the1 for the first ∫ come from and the equal sign=
dq not dt for the first ∫
not used to having q be the variable of integration, as well as one of the limits. care to elaborate?
To understand how the integral expression changes from
(0 to q) ∫ [(q - CE)/rc] dt
to
(0 to q) ∫ [1/(q - CE)] dt = (0 to t) ∫ [1/rc] dt,
let's break it down step by step.
Step 1: Divide the numerator by the denominator
(q - CE) / rc
Step 2: Write the integral symbol ( ∫ ) to represent integration.
(0 to q) ∫ [(q - CE) / rc] dt
Step 3: Rewrite the expression inside the integral using partial fraction decomposition, which means expressing the fraction as a sum of simpler fractions.
(q - CE) / rc = 1 / rc - CE / rc
Step 4: Rewrite the integral with the new partial fraction decomposition.
(0 to q) ∫ [(1 / rc) - (CE / rc)] dt
Step 5: Distribute the integral to both terms inside the brackets.
(0 to q) ∫ (1 / rc) dt - (0 to q) ∫ (CE / rc) dt
Step 6: Integrate each term separately.
(1 / rc) * (t - 0) - (CE / rc) * (t - 0)
Simplifying further:
(1 / rc) * t - (CE / rc) * t
Step 7: Rewrite the integral limits using variable t instead of q.
(0 to t) ∫ (1 / rc) dt - (0 to t) ∫ (CE / rc) dt
Now, you can see how the expression becomes
(0 to t) ∫ (1 / rc) dt = (0 to t) ∫ (1 / rc) dt.
The 1 in the first integral comes from integrating (1 / rc) with respect to t. As a result, it becomes t/(rc), which is why the first integral has a t in the expression.
The equal sign (=) simply represents that both integrals have the same limits of integration, which is from 0 to t.