Antiderivatives
An antiderivative of f is a function F whose derivative is f. Because constants vanish when differentiated, every antiderivative comes with an unknown constant, and the whole family is the indefinite integral: the integral of f(x) dx = F(x) + C.
Basic formulas
- The integral of x^n is x^(n+1)/(n + 1) + C for n not equal to -1. Add one to the power, then divide by the new power.
- The integral of 1/x is ln|x| + C.
- The integral of e^x is e^x + C.
- The integral of cos(x) is sin(x) + C and the integral of sin(x) is -cos(x) + C.
- For a linear inside, such as e^(kx), divide by k: the integral is e^(kx)/k + C.
You can always check an antiderivative by differentiating it. If you do not get the original function back, something is wrong.
The definite integral
The definite integral of f from a to b is the signed area between the graph and the x-axis: area above the axis counts positive, and area below counts negative. It can be built from Riemann sums. Split [a, b] into n strips of width w = (b - a)/n, choose a sample height in each strip, and add up height times width. A left sum uses the left end of each strip and a right sum uses the right end. As n grows, the sums approach the integral.
Properties: integrals add over adjacent intervals, constants factor out, and swapping the limits changes the sign.
Worked example 1
Evaluate the integral of 2x + 1 from 0 to 3. An antiderivative is x^2 + x. At 3 it is 12 and at 0 it is 0, so the answer is 12.
Worked example 2: Riemann sum
Estimate the integral of x from 0 to 2 using a right sum with 4 strips. Each width is 0.5 and the right ends are 0.5, 1, 1.5 and 2. The sum is 0.5(0.5 + 1 + 1.5 + 2) = 2.5. The exact value is 2, so the right sum overestimates, as expected for an increasing function.
Area between curves
For two curves with f above g on [a, b], the area between them is the integral of f(x) - g(x). For y = x and y = x^2 on [0, 1], this gives (1/2 - 1/3) = 1/6.
Displacement and distance
Integrating velocity gives net displacement. For total distance, integrate the absolute value of velocity, splitting at times when velocity changes sign.
Common mistakes
- Forgetting + C on an indefinite integral.
- Using the power rule on 1/x.
- Forgetting to divide by the inner coefficient.
- Finding net displacement when total distance was asked.