Sample Problems. Practice Problems

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1 Lectre Notes Trigonometric Integrals age Samle Problems Comte each of the following integrals.. sin Assme that a an b are ositive nmbers. 8. csc 5. sec ( ). 3. cos 5 cos sin 9.. sin sin cos. csc. sin 7. = cos 5. tan. sin 5 8. tan 3 6. cot 3. a + b 9. sin 7 cos 3 7. sec. a. sin sin Practice Problems.. 3. cos 3 sin 5 sec tan cos () cos 3 cos =6 cos 6 sin a cos 8a a sec tan cot ( ).. 3. cos 5 sin cos 5 sin 3 cos cos b cos b b sin 6 sin 7. cos. tan 9. cos m sin 3m m c coyright Hiegkti, Powell, Last revise: December 8, 3

2 Lectre Notes Trigonometric Integrals age Samle Problems - Answers.) cos + C.) 5 sin 5 + C 3.) 5 sin5 + C.) cot + C 5.) ln jcos j + C = ln jsec j + C 6.) ln jsin j + C 7.) ln jsec + tan j + C 8.) ln jcsc + cot j + C 9.).) 3.) 3 cos3 cos + C.) ab tan sin + 3 sin C.) cos + 3 cos3 sin + C b a + C.) sin + C 5.) ln jsec ( ) + tan ( )j + C 6.) a 5 cos5 + C 6 7.) 8.) sec +ln jcos j+c 9.) cos 8 cos +C.) sin 6 sin +C 8.) 6.).) 3 sin 3 + C.) cos Practice Problems - Answers + C 3.) sec + C.) tan + C 5.) 5 ln sec + C ln jsin ( )j + C 7.) + sin + C 8.) + 8 sin + C 9.) sin 3 sin3 + C sin + 3 sin + C.) 5 sin5 3 sin3 + sin + C.) 6 cos6 + C 3.) 7.) 6 cos6 + 8 cos8 + C.) + tan + C 5.) 3 6.) cos 6a cos a + C sin b + sin b + C 8.) 6 sin 8 sin + C 9.) 6 cos 8m cos m + C 8 c coyright Hiegkti, Powell, Last revise: December 8, 3

3 Lectre Notes Trigonometric Integrals age 3 Samle Problems - Soltions. sin. 3. This is a basic integral we know from i erentiating basic trigonometric fnctions. Since cos = sin, clearly ( cos ) = sin an so sin = cos + C. cos 5 an so 5 =. We know that cos = sin + C. We will se sbstittion. Let = 5 an then = 5 cos 5 = cos = 5 5 cos = sin 5 + C 5 Note: Once we have enogh ractice, there is no nee to erform this sbstittion in writing. We can jst simly write cos 5 = sin 5 + C. 5 cos sin Let = sin. Then = cos. cos sin = sin (cos ) = = C = 5 sin5 + C. csc We nee to remember that cot = csc. csc = csc = cot + C 5. tan Let = cos. Then = sin. sin tan = cos = (sin ) = ( ) = = ln (cos ) + C = ln jsec j + C = ln jj + C = ln jcos j + C 6. cot Let = sin. Then = cos. cos cot = sin = (cos ) = = ln jj + C = ln jsin j + C c coyright Hiegkti, Powell, Last revise: December 8, 3

4 Lectre Notes Trigonometric Integrals age sec sec = sec From here we will se sbstittion. sec + tan. sec + sec tan sec + tan csc sec + tan sec sec + tan = + sec tan sec + tan Recall that Then = sec tan + sec. csc = = sec = sec tan an sec + sec tan = csc From here we will se sbstittion. = csc + cot. Then = csc csc cot. tan = sec. Let = = ln jj + C = ln jsec + tan j + C csc + cot csc csc + cot = + csc cot csc + cot Recall that csc = csc cot an cot = csc. Let csc + csc cot csc + cot = csc + csc cot = ( ) = = ln jj + C = ln jcsc + cot j + C 9. sin Recall the oble angle formla for cosine, cos = sin. We solve this for sin sin = ( cos ) sin = ( cos ) = cos = + C sin + C = sin + C. sin 3 sin 3 = sin sin = sin cos Let = cos : Then = sin sin 3 = sin cos = cos (sin ) = ( ) = = C = 3 cos3 cos + C c coyright Hiegkti, Powell, Last revise: December 8, 3

5 Lectre Notes Trigonometric Integrals age 5. sin We se the oble angle formla for cosine to eress sin. cos = sin =) sin = ( cos ) sin = sin = ( cos ) = ( cos ) = cos + cos We se the oble angle formla for cosine again to eress cos. cos = cos =) cos = (cos + ). sin 5 sin = cos + cos = cos + (cos + ) = cos + 8 cos + = 8 cos + 8 cos = sin + sin C = sin + 3 sin C This metho works with o owers of sin or cos. We will searate one factor of sin from the rest which will be eresse in terms of cos. sin 5 = sin sin = sin sin = sin sin = sin cos = sin cos + cos 3. We rocee with sbstittion. Let = cos : Then = sin. sin 5 = sin cos + cos = cos + cos (sin ) = + ( ) = + = C = cos + 3 cos3 5 cos5 + C a + b The basic integral here is + = tan + C. a = b. This wol be convenient becase then a + b = b + b = b + We nee a sbstittion ner which c coyright Hiegkti, Powell, Last revise: December 8, 3

6 Lectre Notes Trigonometric Integrals age 6 So we will rse this sbstittion. We solve a = b for a ossible vale of an obtain = a. Then b = a an so b b a + b = =. a = b + b ab tan b a b a = + C b + b a = b ab + = ab tan + C. a = a. The basic integral here is This wol be sefl becase then = sin + C. We nee a sbstittion ner which a = a a = a ( ) = a So we will rse this sbstittion. We solve = a for a ossible vale of an obtain = a an = a. a = a a (a) = a a = = sin +C = sin + C a 5. sec ( ) Let =. Then =. sec ( ) sec ( ) = = = ln jsec ( ) + tan ( )j + C sec = sec = ln jsec + tan j + C 6. + cos We will yet again se the oble angle formla for cosine, this time to eliminate the sqare root. cos = cos =) cos = cos + + cos = Since f () = cos is ositive on cos = cos = h ; i ; we can simlify jcos j = cos 3 jcos j jcos j = cos = sin! = sin 3 sin = 3! = 6 c coyright Hiegkti, Powell, Last revise: December 8, 3

7 Lectre Notes Trigonometric Integrals age 7 7. = cos We sbstitte = cos = sin =) sin = cos into this an obtain sin = cos = = cos = Since f () = sin is non-negative on h ; r sin = i ; we can simlify sin = sin = sin 8. tan 3 = sin = cos! = = cos! = = cos =! = + = cos 9. Let = cos. Then = sin sin tan 3 3 = cos 3 = sin sin cos cos 3 = cos 3 sin = 3 ( ) = 3 = 3 3 = 3 = ln jj + C = ln jj + + C sin 7 cos 3 = ln jcos j + sec + C We wil se the roct-to-sm ientities to trasform this roct into a sm. We write the sine formla for the sm an the i erence of these two angles. We will a the two eqations sin = sin (7 + 3) = sin 7 cos 3 + cos 7 sin 3 sin = sin (7 3) = sin 7 cos 3 cos 7 sin 3 sin + sin = sin 7 cos 3 (sin + sin ) = sin 7 cos 3 We can very easily integrate (sin + sin ) sin 7 cos 3 = (sin + sin ) = = ( cos ) + ( cos ) sin + sin + C = cos cos + C 8 c coyright Hiegkti, Powell, Last revise: December 8, 3

8 Lectre Notes Trigonometric Integrals age 8. sin sin We wil se the roct-to-sm ientities to trasform this roct into a sm. We write the cosine formla for the sm an the i erence of these two angles. cos = cos ( + ) = cos cos sin sin cos 6 = cos ( ) = cos cos + sin sin We will sbtract the rst eqation from the rst one cos 6 cos = sin sin (cos 6 cos ) = sin sin We can very easily integrate (cos 6 cos ) sin sin = (cos 6 cos ) = cos 6 cos = (sin 6) (sin ) + C = 6 sin 6 sin + C 8 For more ocments like this, visit or age at htt:// an click on Lectre Notes. qestions or comments to mhiegkti@ccc.e. c coyright Hiegkti, Powell, Last revise: December 8, 3