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Question

If In=tannxdx,then I0+I1+2(I2+...I8)+I9+I10, is equal to

A
9n=1tannxn
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B
1+8n=1tannxn
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C
9n=1tannxn+1
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D
10n=2tannxn+1
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Solution

The correct option is A 9n=1tannxn
We have In=tannxdx=tann2x(sec2x1)dx=tann1xn1In2i.e.In2+In=tann1xn1(n2)
Thus, we have
I0+I1+2(I2+....+I8)+I9+I10=(I0+I2)+(I1+I3)+(I2+I4)+(I3+I5)+(I4+I6)+(I5+I7)+(I6+I8)+(I7+I9)+(I8+I10).=10n=2tann1xn1=9n=1tannxn

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