PROBLEMS ON INTEGRATING TRIG FUNCTIONS

Problem 1 :

∛t2 - sin t dt

Solution :

Given, ∛t2 - sin t dt= ∛t2 dt - sin t dt= t23 dt - sin t dt= t23 + 123 + 1 - (-cos t) + C= t5353 + cos t + C= 35 t53 + cos t + C

Problem 2 :

sec tcos t dt

Solution :

Given, sec tcos t dt= 1cos tcos t dt= 1cos t · 1cos t dt= sec t · sec t dt= sec2t dt= tan t + C

Problem 3 :

1sin2t dt

Solution :

Given, 1sin2t dt= csc2 t dt= -cot t + C

Problem 4 :

(csc v cot v sec v) dv

Solution :

Given, (csc v cot v sec v) dv= 1sin v ⋅ cos vsin v ⋅ 1cos v dv= 1sin v ⋅ 1sin v dv= 1sin2 v dv= csc2 v dv= -cot v + C

Problem 5 :

4 + 4 tan2 v dv

Solution :

Given, 4 + 4 tan2 v dv= 4 + 4 sec2 v - 1 dv= 4 + 4 sec2 v - 4 dv= 4 sec2 v dv= 4 sec2 v dv= 4 tan v + C

Problem 6 :

sec w sin wcos w dw

Solution :

Given, sec w sin wcos w dw= 1cos w ·sin w cos w dw = sin wcos w cos w dw= sin wcos w · 1cos w dw= sin wcos2 w dwu = cos w, du = -sin w dwdw = 1-sin w du= sin wu · 1u · 1-sin w du= -1u · 1u du=- 1u2 du= -u-2 du= -u-2 + 1-2 + 1=-u-1-1= --u-1= --cos-1 w= sec w + C

Problem 7 :

csc w cos wsin w dw

Solution :

Given, csc w cos wsin w dw= 1sin w ·cos w sin w dw = cos wsin w sin w dw= cos wsin w · 1sin w dw= cos wsin2w dwu = sin w, du = cos w dwdw = 1cos w du= cos wu2 · 1cos w du= 1u2 du= u-2 du= u-2 + 1-2 + 1=u-1-1= -u-1= -csc w + C

Problem 8 :

1 + cot2z cot zcsc z dz

Solution :

Given, 1 + cot2z cot zcsc z dz= csc2z cot zcsc z dz= csc z cot z dz= -csc z + C

Problem 9 :

tan zcos z dz

Solution :

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