Midterm-Exam Aid

Midterm-Exam Aid

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ E XAMPLE 5. Calculate ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿￿￿￿￿ ￿ ￿￿, for ￿ ￿ ￿ ￿ ￿. ￿ S OLUTION. The strategy is not immediately obvious. There is no clear simplification to be made, no trigonometric functions, no radicals, and no rational functions. We might be able to use integration by parts, using the “1” trick. (Notice that the integrand is continuous for ￿ ￿ ￿ ￿ ￿). We have Then ￿ ￿ ￿￿￿￿￿￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿￿￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿￿￿ ￿￿ 10 ￿ ￿ ￿￿ ￿ WATERLOO SOS E XAM -AID: MATH138 M IDTERM E XAMPLE 6. Calculate ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿￿￿￿￿￿￿ S OLUTION. We can attempt to simplify the integrand, using the double angle formula for sine, to get ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿￿￿￿￿￿￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿￿￿ There is no obvious antiderivative, nor can we immediately use a substitution like ￿ ￿ ￿￿￿￿￿￿ or ￿ ￿ ￿￿￿￿￿￿ since we do not have an extra term to account for the change of variables. However, notice that we can get a ￿￿￿￿￿￿ term appearing in the denominator if we divide through by ￿￿￿￿ ￿, yielding a ￿￿￿￿ ￿ term in the numerator. We then save a copy of ￿￿￿￿ ￿, and express the remainder in terms of ￿￿￿ ￿: ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿￿￿￿￿￿￿ ￿ ￿￿￿￿ ￿￿￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿ ￿ ￿￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿￿ ￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿￿ ￿ From here it is clear that a good substitution would be ￿ ￿ ￿￿￿ ￿, with ￿￿ ￿ ￿￿￿￿ ￿ ￿￿. Then ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿￿ We can use a partial fraction decomposition now, say ￿ ￿￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿￿ which, after multiplying by ￿￿￿ ￿ ￿ ￿ ￿￿￿ , yields ￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ Comparing coefficients gives ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ Solving, ￿ ￿ ￿￿ Then, ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ 11 ￿ ￿ ￿￿ ￿￿￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿ WATERLOO SOS E XAM -AID: MATH138 M IDTERM For the first integral, we complete the square and make a shift ￿ ￿ ￿ ￿ ￿ : ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿...
View Full Document

This document was uploaded on 03/04/2014 for the course MATH 138 at Waterloo.

Ask a homework question - tutors are online