MATH116 Lecture Notes - Lecture 13: In C, Opata Language, Tael

• School: University of Waterloo
• Department: Mathematics
• Course: MATH116
• Professor: Fiona Dunbar

Document Summary

Practice solutions 10: evaluate the following de nite integrals (cid:90) 0. 0 x3 dx (b) du (c) sin2(5x) dx. 1 + x2 dx (e) xh(x 2) + cos( x)h(x 3) dx. Note: in part (e), h(x) is the heaviside function. (cid:90) 1 (cid:90) . 1 u (cid:12)(cid:12)(cid:12)(cid:12)0 du = ln|1 u| 0 sin2(5x) dx = (cid:90) (cid:90) (cid:18) (cid:18) 1 cos(10x) dx x sin(10x) dx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) 12 (cid:90) 5 (cid:90) 2 (cid:90) 5 (e) xh(x 2) + cos( x)h(x 3) dx. Let f (x) = h(x 2) + cos( x)h(x 3), we then have xh(x 2) + cos( x)h(x 3) dx = 1 f (x) dx f (x) dx + (cid:90) 3. 3 f (x) dx (cid:90) 5 (cid:90) 2. Sin(3 ) (cid:90) 3 (cid:12)(cid:12)(cid:12)(cid:12)3 (cid:18) x2 (cid:18)25. A suitable substitution (possibly after some manipula- tion of the integrand) should work in each case. dx (b) dx (c) (cid:90) 1 sin x (cid:90)