Up to this point we have investigated the definite integral of a function over an interval. In particular we have done the following. Approximated integrals using left and right Riemann sums. Defined the limit of the left Riemann sum (LHS) over an interval [a, b] to be the integral of a function f(x) from a to b, denoted b a f(x) dx. Calculated definite integrals numerically using our calculators. Today we will algebraically determine the indefinite integral of an arbitrary function f(x). Antiderivative An antiderivative of a function f is a function F such that F = f. It may not be immediately clear what this definition is describing. Let s consider some examples to develop our understanding of the antiderivative. Example 1 Determine an antiderivative of 2x. This question can be reformulated as the statement below. The derivative of is 2x. So instead of being given a function and finding its derivative, we re given a derivative and we need to find a function it could have been derived from. So we need to think of a function we could differentiate and get 2x. Initially solving these problems will likely take some trial and error. Notice that 2x is a power function whose exponent is 1. When we take the derivative of a power function we lose a power so our antiderivative should have an x 2 term (Why must this be true?). We can check if x 2 works as an antiderivative by simply taking its derivative. If we get 2x then x 2 is an antiderivative of 2x. Since (x 2 ) = 2x, we see that an antiderivative of 2x is x 2. 1 Example 2 Determine an antiderivative of 8x 3. Again, it may help to rephrase this question as the statement below. The derivative of is 8x 3. Since 8x 3 is a power function we expect its antiderivative to be a power function with an exponent of 4, otherwise its derivative won t give us a power function with an exponent 3. So let s check to see if x 4 works as an antiderivative of 8x 3. 1 If we had tried something like x 5 we would have that (x 5 ) = 5x 4 so since 5x 4 2x we see x 5 is not an antiderivative of 2x. 108
Taking the derivative we get (x 4 ) = 4x 3. This is close but we need the coefficient to be 8 so we see by trying 2x 4 (since 2 4 = 8) that an antiderivative of 8x 3 is 2x 4 (since the derivative of 2x 4 is 8x 3 ). Antiderivatives are not unique. That is, given a function and asked to find its antiderivative, you and I might both come up with different antiderivatives. For instance in example 1 we found that an antiderivative of 2x is x 2. However the derivative of a constant is zero, so x 2 +4 is also an antiderivative of 2x since the derivative of x 2 + 4 is 2x. Notice the issue of non-uniqueness arises because the derivative of a constant is zero. We can represent the set of all antiderivatives of 2x by x 2 +C where C is any constant. The expression x 2 +C is known as the indefinite integral of 2x. The Indefinite Integral The integral given by f(x)dx, is known as the indefinite integral. This expression represents the set of all antiderivatives. Example 3 Determine 2x dx. The indefinite integral of 2x is the set of all antiderivatives so we have 2xdx = x 2 + C, where C is some constant 2. We can check our answer by differentiating. To check the example above we check that (x 2 + C) equals 2x (it does!). Example 4 Determine 4x 3 dx. The indefinite integral of 4x 3 is the set of all antiderivatives so we have 4x 3 dx = x 4 + C, where C is some constant. The function f(x) being integrated is called the integrand, and the variable x is called the variable of integration. 2 When used in an indefinite integral we call the constant C the constant of integration. 109
You may have noticed that for the power functions above we can find their integral by taking the reverse derivative rule. In fact, just like derivatives, we have rules for integration. Power Rule for the Indefinite Integral ( ) 1 x n dx = x n+1 + C, for n 1. n + 1 x 1 dx = ln x + C, for n = 1. In both cases both n and C are constants. Remember x 1 = 1 x so our second rule can also be written as 1 dx = ln x + C. x Example 5 Determine x dx. Similar to differentiation, it is helpful to first rewrite the x as a power of x. This gives us xdx = x 1/2 dx Following the rule above we have that n = 1/2 so we get the following indefinite integral. ( ) x 1/2 1 dx = x 1/2+1 + C, 1/2 + 1 = 1 3/2 x3/2 + C, where C is some constant. = 2 3 x3/2 + C. 1. Determine x 42 dx. 110
In addition to the indefinite integral of power functions we have the following rules for the other functions we have been working with. 3 Indefinite Integral of e x, b x, and x e x dx = e x + C, for If b 1 is any positive number then b x dx = bx ln(b) + C, x dx = x x 2 + C We have integral rules for sums, differences, and constant multipliers as well. Sums, Differences, and Constant Multiples [f(x) ] ± g(x) dx = f(x)dx ± g(x)dx If k is any constant, then we have the following rule. kf(x)dx = k f(x)dx Example 6 Use the rules above to determine the indefinite integral given by (10x 4 + 2x 2 3e x ) dx 3 There rules for the indefinite integral of other functions but for our purposes we will only use the rules presented here. 111
Example 7 By the start of 2008, Apple had sold a total of about 3.5 million iphones. From the start of 2008 through the end of 2010, sales of iphones were approximately s(t) = 4.5t 2 + 5.8t + 6.5 million iphones per year (0 t 3), where t is time in years since the start of 2008. Find an expression for the total sales of iphones up to time t. 2. Determine the following indefinite integrals. (a) x 5 dx (b) (1 + u) du (c) 4 x dx (d) (3.2 + 1 ) dt t0.9 (e) ( 6.1 x0.5 + x0.5 6 ex ) dx 112
3. The velocity of a particle moving along a straight line is given by v(t) = 4t + 1 m/sec. given that the particle is at position s = 2 meters at time t = 1, find an expression for s in terms of t. That is, find s(t) and use the given information to determine the constant of integration. 4. Explain why the following solutions were marked incorrect and determine the correct answer. (a) 4(e x 2x) dx= (4x)(e x x 2 ) + C Wrong (b) (2 x 1) dx= 2x+1 x + 1 x + C Wrong 113