# $\int \sec ^{3}\theta d\theta=$ Option 1)             1/2 [tan $\theta$ sec $\theta$ + log |(tan $\theta$+ sec $\theta$)|] + c Option 2) 1/2 [tan $\theta$ + sec $\theta$ - log|(tan $\theta$ + sec $\theta$)|] + c Option 3) 1/2  [tan $\theta$ sec $\theta$ - log |(sec $\theta$ - tan $\theta$)|] + c Option 4) none of these

V Vakul

As we learnt

Integration By PARTS -

Let $u$ and $v$ are two functions then

$\int u\cdot vdx=u\int vdx-\int \left ( \frac{du}{dx}\int vdx \right )dx$

- wherein

 Where $u$ is the Ist function $v$ is he IInd function

I                             = ò sec q sec2q dq

= sec q tan q -ò sec q tan2q dq

= sec q tan q - ò sec3 q dq + ò sec q dq

\   2I = sec q tan q + log |sec q + tan q| + c

Þ  I = 1/2[tan q sec q + log|tan q +  sec q|] + c

Option 1)

1/2 [tan $\theta$ sec $\theta$ + log |(tan $\theta$+ sec $\theta$)|] + c

Option 2)

1/2 [tan $\theta$ + sec $\theta$ - log|(tan $\theta$ + sec $\theta$)|] + c

Option 3)

1/2  [tan $\theta$ sec $\theta$ - log |(sec $\theta$ - tan $\theta$)|] + c

Option 4)

none of these

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