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Question

If In=cotnxdx and I0+I1+2(I2+...+I8)+I9+I10=A(u+u22+...+u99)+ constant where u=cotx then,

A
A is constant
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B
A=1
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C
A=1
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D
A is dependent on x
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Solution

The correct options are
A A is constant
B A=1
In=cotnxdx=cotn2xcot2xdx=cotn2x(csc2x1)dx=cotn2xcsc2xdxIn2
In+In2=cotn1xn1+c
I0+I1+2(I2+...I8)+I9+I10=(I2+I0)+(I3+I1)+(I4+I2)+(I5+I3)+(I6+I4)+(I7+I5)+(I8+I6)+(I9+I7)+(I10+I8)
=(cotx1+cot2x2+...+cot9x9)+c
Hence A=1

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