Integration

• Integration can be used to find areas bounded between a curve and the coordinate axes.
• Integration is the reverse of differentiation, therefore is dy/dx = f(x), y = $\int$ f(x) dx.
• After every integration, a constant, c, must be added as integrating is indefinite.
• A definite integral is one bounded between two limits.
• $\int$ a x ^n dx = a $\int$ x ^n dx = (a/n + 1) x ^(n + 1).
• $\int$ x ^ n + x ^ m dx = $\int$ x ^ n dx + $\int$ x ^ m dx.
• When integrating over an interval that isn't entirely above or below the x axis, split the integral into separate regions.
• $\int$ e ^ a x dx = 1/a e ^ a x.
• $\int$ 1/ a x dx = 1/a ln (a x).
• $\int$ sin a x dx = - 1/a cos a x.
• $\int$ cos a x dx = 1/a sin a x.
• $\int$ u dv/dx dx = u v - $\int$ v du/dx dx.
• $\int$ f'(x) f(x) ^ n dx = f(x)^(n + 1)/n + 1.
• $\int$ f'(x)/f(x) dx = ln f(x).
• The trapezium rule is $\int_a^b$ y dx = 1/2 h (y0 + yn + 2(y1 + y2 +...+ y(n-1))).
• The trapezium rule gives an overestimate when the curve is convex.
• The trapezium rule gives an underestimate when the curve is concave.
• When dy/dx = f(x)g(y), $\int$ 1/g(y) dy = $\int$ f(x) dx.