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CEGEP CHAMPLAIN - ST. LAWRENCE 201-203-RE: Integral Calculus Patrice Camiré Problem Sheet #9 Trigonometric Integrals 1. Evaluate the following trigonometric integrals: sin(x) and/or cos(x) with at least one odd power. 2 Z 2 sec(x) dx = ln | sec(x) + tan(x)| + C sin (x) + cos (x) = 1 d sin(x) = cos(x) dx Z Z sin(x) cos(x) dx (a) Z (b) Z (c) Z (d) Z (e) Z (f) Z (g) d cos(x) = − sin(x) dx (h) Z sin(x) cos2 (x) dx (i) Z sin3 (x) cos2 (x) dx (j) Z cos5 (x) sin2 (x) dx (k) sin8 (x) cos3 (x) dx sin3 (x) dx Z cos3 (x) dx Z sin4 (x) dx cos(x) Z sin3 (4x) dx Z sin3 (2x) cos3 (2x) dx Z cos5 (x) sin3 (x) dx (o) sin(x) dx cos3 (x) (p) cos3 (x) dx sin(x) (q) sin3 (x) p dx cos(x) Z (r) cos5 (x) sin4 (x) dx (s) Z sin3 (x) dx cos(x) Z p 3 (t) cos(x) sin3 (x) dx Z sin2 (x) dx cos(x) (l) (m) cos5 (x) dx cos(x) dx 1 + sin(x) (n) Z (u) sin5 x 2 dx 2. Evaluate the following trigonometric integrals: sin(x) and/or cos(x) with only even powers. cos2 (x) = Z (a) Z (b) sin2 (x) dx 2 cos (x) dx 1 + cos(2x) 2 Z (c) Z (d) sin2 (x) = sin2 (x) cos2 (x) dx 2 sin (3x) dx 1 − cos(2x) 2 Z cos2 (5x) dx Z sin2 (3x + 1) dx (e) (f) 3. Evaluate the following trigonometric integrals: tan(x) and/or sec(x). 2 2 tan (x) + 1 = sec (x) d tan(x) = sec2 (x) dx Z (a) sec (x) dx Z (b) 2 tan (x) dx Z (c) Z (d) sec4 (x) dx 4 tan (x) dx Z (e) Z (f) Z (g) Z (h) 2 tan(2x) sec2 (2x) dx sec(x) tan3 (x) dx tan3 (x) sec2 (x) dx sec(x) tan(x) dx sec(x) − 1 Z Z tan(x) dx = ln | sec(x)| + C sec(x) dx = ln | sec(x) + tan(x)| + C d sec(x) = sec(x) tan(x) dx Z (i) 5 tan (x) dx Z (j) Z (k) (m) Z (n) Z (o) sec6 (x) dx Z tan3 (x) dx sec(x) Z tan2 (x) sec4 (x) dx Z tan3 (x) sec5 (x) dx Z tan5 (x) sec(x) dx Z tan3 (x) dx sec2 (x) Z sec(x) dx tan2 (x) Z sin2 (x) dx cos6 (x) Z sin4 (x) dx cos2 (x) tan3 (x) dx 3 tan5 (x) sec7 (x) dx (s) (t) tan2 (x) dx sec(x) 8 (u) sec (x) tan(x) dx Z (p) Z (r) tan(x) sec (x) dx Z sec4 (x) dx (q) tan8 (x) sec4 (x) dx Z (l) Z 3 tan (x) sec(x) dx 4. Evaluate the following trigonometric integrals: miscellaneous. Z Z 1 − sin(x) tan(x) dx (e) dx (a) sec(x) cos(x) Z Z tan4 (x) sin2 (x) (f) dx (b) dx cos4 (x) cos4 (x) Z Z 1 4 (g) dx (c) tan(x) cos (x) dx 2 sin (x) sec3 (x) √ Z Z tan3 (x) sin3 ( x) √ (d) dx (h) dx cos3 (x) x (v) (i) (j) (k) (l) Answers 1. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) 1 sin2 (x) + C 2 1 − cos3 (x) + C 3 1 1 cos5 (x) − cos3 (x) + C 5 3 2 1 1 3 sin (x) − sin5 (x) + sin7 (x) + C 3 5 7 1 1 sin9 (x) − sin11 (x) + C 9 11 1 cos3 (x) − cos(x) + C 3 2 1 sin(x) − sin3 (x) + sin5 (x) + C 3 5 ln |1 + sin(x)| + C 1 sec2 (x) + C 2 1 ln | sin(x)| − sin2 (x) + C 2 2 cos5/2 (x) − 2 cos1/2 (x) + C 5 x 1 − sin(2x) + C 2 4 x 1 (b) + sin(2x) + C 2 4 2. (a) 1 2 1 sin5 (x) − sin7 (x) + sin9 (x) + C 5 7 9 cos2 (x) (m) − ln | cos(x)| + C 2 (n) ln | sec(x) + tan(x)| − sin(x) + C (l) (o) sin(x) − (p) ln | sec(x) + tan(x)| − sin(x) − (q) (r) (s) (t) (u) sin3 (x) +C 3 cos3 (4x) cos(4x) − +C 12 4 sin4 (2x) sin6 (2x) − +C 8 12 cos8 (x) cos6 (x) − +C 8 6 3 3 cos10/3 (x) − cos4/3 (x) + C 10 4 x 4 x 2 x −2 cos + cos3 − cos5 +C 2 3 2 5 2 1 x − sin(4x) + C 8 32 x sin(6x) (d) − +C 2 12 (c) 3. (a) tan(x) + C (b) tan(x) − x + C 1 (c) tan(x) + tan3 (x) + C 3 1 3 (d) tan (x) − tan(x) + x + C 3 1 (e) tan2 (2x) + C 4 1 (f) sec3 (x) − sec(x) + C 3 1 (g) tan4 (x) + C 4 (h) ln | sec(x) − 1| + C 1 1 (i) tan4 (x) − tan2 (x) + ln | sec(x)| + C 4 2 1 1 (j) tan9 (x) + tan11 (x) + C 9 11 1 (k) tan2 (x) − ln | sec(x)| + C 2 1 (l) sec3 (x) + C 3 sin3 (x) +C 3 x sin(10x) + +C 2 20 x sin(6x + 2) (f) − +C 2 12 (e) 1 2 1 sec11 (x) − sec9 (x) + sec7 (x) + C 11 9 7 (n) ln | sec(x) + tan(x)| − sin(x) + C (m) sec8 (x) +C 8 sec3 (x) − sec(x) + C (p) 3 tan3 (x) (q) tan(x) + +C 3 tan5 (x) 2 (r) + tan3 (x) + tan(x) + C 5 3 (s) sec(x) + cos(x) + C (o) tan3 (x) tan5 (x) + +C 3 5 sec7 (x) sec5 (x) (u) − +C 7 5 sec5 (x) 2 (v) − sec3 (x) + sec(x) + C 5 3 (t) 4. (a) − cos(x) + C 1 (b) tan3 (x) + C 3 1 (c) − cos4 (x) + C 4 1 1 (d) − sec3 (x) + sec5 (x) + C 3 5 (e) ln |1 + sin(x)| + C 1 1 (f) tan5 (x) + tan7 (x) + C 5 7 (g) − csc(x) − sin(x) + C (h) √ √ 2 cos3 ( x) − 2 cos( x) + C 3 cos2 (x) − ln | cos(x)| + C 2 1 (j) − +C sin(x) (i) (k) tan3 (x) tan5 (x) + +C 3 5 3 sin(2x) (l) tan(x) − x − +C 2 4