Integration by parts formula

\int u d v = u v - \int v d u

Integration by parts for definite integrals

\int_{a}^{b} u d v = {u v |}_{a}^{b} - \int_{a}^{b} v d u

To integrate products involving $sin (a x),$ $sin (b x),$ $cos (a x),$ and $cos (b x),$ use the substitutions.

Sine Products

sin (a x) sin (b x) = \frac{1}{2} cos ((a - b) x) - \frac{1}{2} cos ((a + b) x)

Sine and Cosine Products

sin (a x) cos (b x) = \frac{1}{2} sin ((a - b) x) + \frac{1}{2} sin ((a + b) x)

Cosine Products

cos (a x) cos (b x) = \frac{1}{2} cos ((a - b) x) + \frac{1}{2} cos ((a + b) x)

Power Reduction Formula

\int {sec}^{n} x d x = \frac{\sec^{n - 2} x \tan x}{n - 1} + \frac{n - 2}{n - 1} \int {sec}^{n - 2} x d x; n \neq 1

Power Reduction Formula

\int {tan}^{n} x d x = \frac{1}{n - 1} {tan}^{n - 1} x - \int {tan}^{n - 2} x d x

Midpoint rule

M_{n} = \sum_{i = 1}^{n} f (m_{i}) Δ x

Trapezoidal rule

T_{n} = \frac{1}{2} Δ x (f (x_{0}) + 2 f (x_{1}) + 2 f (x_{2}) + \dots + 2 f (x_{n - 1}) + f (x_{n}))

Simpson’s rule

S_{n} = \frac{Δ x}{3} (f (x_{0}) + 4 f (x_{1}) + 2 f (x_{2}) + 4 f (x_{3}) + 2 f (x_{4}) + 4 f (x_{5}) + \dots + 2 f (x_{n - 2}) + 4 f (x_{n - 1}) + f (x_{n}))

Error bound for midpoint rule

Error in M_{n} \leq \frac{M {(b - a)}^{3}}{24 n^{2}}

Error bound for trapezoidal rule

Error in T_{n} \leq \frac{M {(b - a)}^{3}}{12 n^{2}}

Error bound for Simpson’s rule

Error in S_{n} \leq \frac{M {(b - a)}^{5}}{180 n^{4}}

Improper integrals

\begin{matrix} \int_{a}^{+ \infty} f (x) d x = lim_{t \to + \infty} \int_{a}^{t} f (x) d x \\ \int_{− \infty}^{b} f (x) d x = lim_{t \to − \infty} \int_{t}^{b} f (x) d x \\ \int_{− \infty}^{+ \infty} f (x) d x = \int_{− \infty}^{0} f (x) d x + \int_{0}^{+ \infty} f (x) d x \end{matrix}