Calculus Reference
Antiderivatives and the Fundamental Theorem
Integration reverses differentiation and measures accumulated change, area, and net change.
Solved Examples
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Core Ideas
Indefinite Integral
$$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$$
Fundamental Theorem
$$\int_a^b f(x)\,dx=F(b)-F(a)$$
The constant $C$ is required for an indefinite integral because every antiderivative differs by a constant.
Find an Antiderivative
Power Expression
Find an antiderivative of the expression $f(x)=3x^2-4x+1$.
Solution Steps
- Increase each power by $1$ and divide by the new power.
$$\int(3x^2-4x+1)\,dx=x^3-2x^2+x+C$$
Evaluate a Definite Integral
Use the Fundamental Theorem
Evaluate the definite integral $\int_0^2(3x^2-4x+1)\,dx$.
Solution Steps
- Find an antiderivative.
- Subtract its value at the lower bound from its value at the upper bound.
- State the definite integral: $2$.
$$F(x)=x^3-2x^2+x$$$$\int_0^2(3x^2-4x+1)\,dx=F(2)-F(0)$$$$=(8-8+2)-0=2$$
Use an Accumulation Function
Differentiate an Integral
Let $A(x)=\int_1^x(t^2+1)\,dt$. Find $A'(x)$.
Solution Steps
- Apply the Fundamental Theorem to the moving upper bound.
$$A'(x)=x^2+1$$