If
P'(t)= [10(t+2)]/t. What is P(t)?
It is asking for the derivative, and I am so lost! Because so far we haven't covered how to find the derivative of something so complicated. Any ideas? TY
So I meant it is asking for the ANTIderivative
This is not complicated.
P'(t)=10 + 20/t
integral P'dt= 10t +20ln(t)
To find P(t) given the derivative P'(t), we will use the process of integration. Integration is the reverse process of differentiation.
Given P'(t) = [10(t+2)]/t, we can integrate both sides with respect to t to find P(t).
∫P'(t) dt = ∫[10(t+2)]/t dt
To solve this integral, we need to split the expression into two parts:
∫[10(t+2)]/t dt = ∫(10/t)(t+2) dt
We can separate this using the distributive property:
∫(10/t)(t+2) dt = ∫(10/t)*t dt + ∫(10/t)*2 dt
Simplifying further:
= 10 ∫dt + 20 ∫(1/t)dt
The integral of dt is simply t, and the integral of (1/t)dt is ln|t| (natural logarithm of the absolute value of t).
Therefore, our expression becomes:
= 10t + 20ln|t| + C
where C is the constant of integration.
Hence, P(t) = 10t + 20ln|t| + C, where C is a constant.