Integration (Pure 2)
Standard integrals
$$\int e^{ax+b}\,dx = \frac{1}{a}e^{ax+b} + C, \qquad \int \frac{1}{ax+b}\,dx = \frac{1}{a}\ln|ax+b| + C,$$
$$\int \cos(ax+b)\,dx = \frac{1}{a}\sin(ax+b) + C, \qquad \int \sin(ax+b)\,dx = -\frac{1}{a}\cos(ax+b) + C.$$
- To integrate a power of $\sin$/$\cos$, use an identity to remove the power first.
Practice
What is ∫ eˣ dx?
eˣ is its own integral (and derivative): ∫eˣ dx = eˣ + C.
Practice
What is ∫ (1/x) dx?
∫(1/x) dx = ln|x| + C (the power rule fails for n = −1).
The trapezium rule
- When you can't integrate exactly, estimate:
$$\int_a^b y\,dx \approx \tfrac{h}{2}\big[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\big]$$
- Each strip of width $h$ is a trapezium; add their areas.
Practice
The trapezium rule estimates a definite integral when it cannot be found exactly.
It approximates the area as a sum of trapezium strips of width h.
You've got it
Key idea
- $\int e^{ax+b}dx = \tfrac1a e^{ax+b} + C$; $\int \tfrac{1}{ax+b}dx = \tfrac1a\ln|ax+b| + C$
- remove powers of $\sin$/$\cos$ with an identity before integrating
- the trapezium rule estimates an integral as a sum of trapezium strips