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Question

The intergral sec2x(secx+tanx)92dx equal (for some arbitrary constant k)

A

1(secx+tanx)112{11117(secx+tanx)2}+K

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B
1(secx+tanx)112{11117(secx+tanx)2}+k
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C
1(secx+tanx)112{111+17(secx+tanx)2}+k
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D
1(sexx+tanx)112{111+17(secx+tanx)2}+k
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Solution

The correct option is C 1(secx+tanx)112{111+17(secx+tanx)2}+k
I=sec2x(secx+tanx)9/2dxLet secx+tanx=tsec xtan x=1tsecx=12(t+1t)Also secx(sec x+tan x)dx=dtsec xdx=dttI=12(t+1t)dtt92.t=12(t9/2+t13/2)dt=12[t9/2+192+1+t13/2+1132+1]+K=17t7/2111t112+k=17t7/2111t11/2+K=1t11/2(111+t27)+K=1(secx+tanx)11/2{111+17(secx+tan x)2}+K

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