To integrate x cos x between the limits x = 3 and x = 5, we first choose u = x and v' = cos(x) and work through the details of integration by parts.

• As we obtain the different terms in our antiderivative, we can apply the limits x = 3 and x = 5 to each in the appropriate manner, as shown below.
• As an alternative we could have found an antiderivative of x cos x, using integration by parts, and then applied the limits to this antiderivative.
• The difference between the two choices is a matter of taste, and is in any case minor.

Using practically the same procedure as that used to find the integral of cos^2(`theta) in the preceding class, we obtain the result in the first line below for the integral of sin^2(`theta).

• Since sin^2(`theta) = (1 - cos(2`theta)) / 2, we should get the same result by integrating this expression.
• Using the substitution u = 2 `theta we easily obtain the result in the third line below.
• Using the trigonometric identity for sin(2 `theta), we see that this result is identical to our original.

To 'normalize' the quantum mechanical wave function sin^2(n `pi x), we find the constant C1 that makes the integral of C1 sin^2(n `pi x) between limits x = 0 and x = 1 equal to 1.

• We will evaluate the integral using the result obtained earlier for the antiderivative of sin^2(x).
• We make the substitution u = n `pi x, as shown below.
• In this case it is convenient to transform the limits of the integral to limits on u: when x = 0 we have u = 0, and x = 1 implies u = n `pi.
• We evaluate our antiderivative at the limits u = 0 and u = n `pi.
• At both of these limits sin u is 0, so the first term of the antiderivative contributes nothing to the integral.
• We obtain the results shown in the last two lines below.
• Our conclusion is that the wave function is normalized for C1 = `sqrt(2).