Goal: determine the solution $F(t)$.
$F(t)$: the antiderivative of $f(t)$.
A function has one derivative.
A function has many antiderivatives:
$F(t) + C$ for any constant number $C$
Example: \begin{align*} \diff{F}{t} = 4t \end{align*}
Antiderivative: `{}`
$$\int 4t\,dt =2 t^2 + C$$
Power rule for derivatives: $\displaystyle \diff{t^{n+1}}{t} = (n+1)t^n$ $\Rightarrow$ $\displaystyle \diff{}{t}\left(\frac{t^{n+1}}{n+1}\right)=t^n$
Example:
$$\int t^2 dt = \frac{t^{2+1}}{2+1}+C = \frac{t^3}{3} + C$$
Back to first example: $\displaystyle \diff{F}{t} = 4t$
$\displaystyle F(t) = \int 4t \, dt$ $\displaystyle = 4 \int t \, dt$ $\displaystyle = 4 \int t^1 \, dt$
$\displaystyle = 4 \frac{t^2}{2} +C = 2t^2 +C$
Example: $\displaystyle \int (2x^3-4x+1)dx$
$\displaystyle = 2 \int x^3dx - 4 \int x\,dx + \int 1 \, dx$
$\displaystyle = 2 \int x^3dx - 4 \int x\,dx + \int x^0 \, dx$
$\displaystyle = 2 \frac{x^4}{4} - 4 \frac{x^2}{2} + \frac{x^1}{1} + C$ $\displaystyle = x^4/2 - 2x^2 + x + C$
Recall derivative:
$\displaystyle \diff{}{t} e^t = e^t$
Reverse for integration:
$\displaystyle \int e^t dt = e^t +C$
$\displaystyle \diff{}{t} e^{3t} = 3 e^{3t}$
$\displaystyle \int 3 e^{3t} dt = e^{3t} +C$
$\displaystyle \diff{}{t} e^{kt} = k e^{kt}$
$\displaystyle \int k e^{kt} = e^{kt} +C$
More examples:
$\displaystyle \int (3+ 10e^{5t}) dt$ $\displaystyle = \int 3 dt + 2\int 5e^{5t} dt$ $\displaystyle =3t + 2e^{5t} +C$
$\displaystyle\int e^{7x} dx$ $\displaystyle = \frac{1}{7}\int 7e^{7x} dx$ $\displaystyle = \frac{1}{7} e^{7x} +C$
Power rule for $n \ne -1$:
$\displaystyle \int t^n dt = \frac{t^{n+1}}{n+1} + C$.
What about $n=-1$? $\displaystyle \int \frac{1}{t} dt = ?$
Recall derivative of logarithm:
$\displaystyle \diff{}{t} \ln(t) = \frac{1}{t}$
Reverse: $\displaystyle \int \frac{1}{t} dt = \ln(t) + C$
Logarithm: defined only for $t > 0$.
Integral: symmetric for $t<0$.
Add absolute value to fix rule: