(i) If In=∫secn⁡xdx then show that In=secn−2⁡xtan⁡xn−1+n−2n−1In−2 Hence find ∫sec5⁡xdx(ii) If In=∫tann⁡xdx then In=tann−1⁡xn−1−In−2 and hence evaluate ∫tan6⁡xdx iii) If In=∫cosecn⁡xdx thenIn=−cosecn−2⁡xcot⁡xn−1+n−2n−1In−2

Question

(i) If In=∫secn⁡xdx then show that In=secn−2⁡xtan⁡xn−1+n−2n−1In−2 Hence find ∫sec5⁡xdx(ii) If In=∫tann⁡xdx then In=tann−1⁡xn−1−In−2 and hence evaluate ∫tan6⁡xdx iii) If In=∫cosecn⁡xdx thenIn=−cosecn−2⁡xcot⁡xn−1+n−2n−1In−2

Accepted Answer

(i) In=∫secn⁡xdx=∫secn−2⁡xsec2⁡xdx=secn−2⁡xtan⁡x−∫tan⁡x(n−2)secn−3x⁡sec⁡xtan⁡xdx=secn−2⁡xtan⁡x-(n−2)∫secn−3⁡sec⁡xtan2⁡xdx=secn−2⁡xtan⁡x−(n−2)∫secn−2⁡xtan2⁡xdx secn−2⁡xtan⁡x−(n−2)∫secn⁡xdx+(n−2)∫secn−2⁡xdx=secn−2⁡xtan⁡x−(n−2)In+(n−2)In−2In(1+n−2)=secn−2⁡xtan⁡x+(n−2)In−2∴In=secn−2⁡xtan⁡xn−1+n−2n−1In−2I5=∫sec5⁡xdx=sec3⁡xtan⁡x4+34I3=sec3⁡xtan⁡x4+34sec⁡xtan⁡x2+12I1∴I5=sec3⁡xtan⁡x4+38sec⁡xtan⁡x+38log⁡|sec⁡x+tan⁡x|+c  (ii) In=∫tann−2⁡xtan2⁡xdxIn=∫tann−2⁡xsec2⁡x−1dx=∫tann−2⁡xsec2⁡xdx−∫tann−2⁡xdxIn=tann−1⁡xn−1−In−2I6=∫tan6⁡xdx=tan5x5−I4=tan5⁡x5−tan3⁡x3−I2=tan5⁡x5−tan3⁡x3+tan⁡x−I0∴I6=tan5⁡x5−tan3⁡x3+tan⁡x−x+c   (iii) In=∫cosecn⁡xdx=∫cosecn−2⁡xcosec2⁡xdx=cosecn−2⁡x(−cot⁡x)-∫cot⁡x(n−2)cosecn−3⁡x(cosecxcot⁡x)dx =−cosecn−2⁡xcot⁡x-(n−2)∫cosecn−2⁡xcosec2⁡x−1dx=−cosecn−2⁡xcot⁡x+(n−2)In−2−(n−2)InIn(1+n−2)=−cosec2⁡x−2xcot⁡x+(n−2)In−2 In=−cosecn−2⁡xcot⁡xn−1+n−2n−1In−2n=5⇒I5=−cosec3⁡xcot⁡x4+34I3                    =−cosec3⁡xcot⁡x4+34(−cosec⁡xcot⁡x2+12I1)=−cosec3⁡xcot⁡x4−38cosec⁡xcot⁡x+38I1=−cosec3⁡xcot⁡x4−38cosec⁡xcot⁡x+38logcosecx-cotx+c

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