If I n = ∫ 0 π 4 tan n x d x , then 1 I 2 + 1 4 + 1 I 3 + I 5 + 1 I 4 + 1 6 =

If In=0π4tannxdx, then 1I2+14+1I3+I5+1I4+16=
  1. 1I9+I11
  2. 1I10+I12
  3. 1I12+I14
  4. 1I11+I13

Solution

Given In=0π4tannxdx

In+In+2=0π4tannxdx+0π4tann+2xdx

=0π4tannxsec2xdx

=01tndt where tanx=t

=tn+1n+101=1n+1

Now, 1I2+14+1I3+I5+1I4+16=3+4+5=12

=1I11+I13

Asked in: AP EAMCET 2022 (04 Jul Shift 2)

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