Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 7]_Integrating Trigonometric Functions_131
2. Integrals of the form V sin mx cos nxdx, \ sin rnx sin nx dx and
V cos mx cos nx dx. In these cases the following formulas are used;
1) sin mx cos nx = -~ [sin (m + n) x + sin (m n) x] ;
2) sin mx sin nx = -^ [cos (m n) x cos(m + n) x]\
3) cosm*cos ^^ = -9- [ c s (m n)*-fcos (m-f/i) x].
Example
7. I sin 9* sin xdx= \
-^ [cos 8* cos lOjt] dx=*
Find the integrals:
1365.
1366.
^ sin 3jc cosSxdx. 1369. J cos(aA: f
J sin 10* sin 15 A: d*. 1370. $ sin a>/ sin (co/ -f-cp) dt.
1367. f cos ~ cos ^- djc, 1371. ^ cos x cos* 3x dx .
1368. f sin |- sin ~ djc. 1372. J sin x sin 2* sin 3* dx.
3. Integrals
of the form
where R is a rational function.
1) By means of substitution
whence
2*
f /? (sin *, cos x) dx, (2)
x
.
tan t,
integrals of form (2) are reduced to integrals of rational functions by the
new variable t.
Example 8. Find
f_*E_ /.
J 1 + sin x + cos x
Solution. Putting tan -pr^*' we wil1 have
f*
J