Integration and area under a curve

What you need to know

• Integration is the reverse of differentiation, so the integral of ax^n is a x^(n+1)/(n + 1) + c. Add one to the power, then divide by the new power. This is exactly the opposite of what you do when differentiating.
• The rule fails when n = -1, because dividing by zero is impossible. Instead, the integral of 1/x is ln x + c, and this special case is examined often enough that forgetting it is expensive.
• Every indefinite integral must carry the arbitrary constant c. If the question gives a boundary condition, such as the curve passing through a named point, substitute it to find c and then write the particular equation.
• The standard integrals are: integral of sin x dx = -cos x + c, integral of cos x dx = sin x + c, integral of sec^2 x dx = tan x + c and integral of e^x dx = e^x + c. Note the minus sign on the cosine, which is the reverse of the differentiation pattern.
• For a definite integral, find the antiderivative, write it in square brackets with the limits, substitute the upper limit, then subtract the value at the lower limit. There is no constant of integration in a definite integral, since it cancels in the subtraction.
• Integration by substitution handles a function of a linear function: the integral of (ax + b)^n dx is (ax + b)^(n+1)/(a(n + 1)) + c. The extra division by a is the step candidates omit, so check by differentiating your answer back.
• For a general substitution, let u be the inner function, find du/dx, replace dx completely, and change the limits to u values if the integral is definite. Never mix u and x in the same integral.
• The area between a curve and the x-axis from x = a to x = b is the definite integral of y dx between those limits. Sketch the curve first so that you know where it crosses the axis.
• A region below the x-axis gives a negative integral. Area is never negative, so split the integral at each root and take the absolute value of each piece before adding, otherwise the parts cancel and you report an area that is too small.
• The area between two curves is the integral of (upper curve minus lower curve) between their points of intersection. Find the intersections first by setting the two equations equal to each other and solving.
• The volume generated when the region under a curve is rotated completely about the x-axis is pi times the integral of y^2 dx. Square y before integrating, not after, and remember that the pi stays outside throughout.
• To integrate a product of trigonometric functions, first use an identity to turn it into a sum. For instance sin^2 x is rewritten as (1 - cos 2x)/2, which you can integrate term by term.
• The trapezium rule gives an approximate value when exact integration is impossible: area is approximately (h/2)[first ordinate + last ordinate + 2(sum of the remaining ordinates)], where h is the common width.