Here you will learn proof of integration of sec inverse x and cosec inverse x.
Let’s begin –
Integration of Sec Inverse x
The integration of sec inverse x or arcsec x is \(xsec^{-1}x\) – \(log|x + \sqrt{x^2 – 1}|\) + C
Where C is the integration constant.
i.e. \(\int\) \(sec^{-1}x\) = \(xsec^{-1}x\) – \(log|x + \sqrt{x^2 – 1}|\) + C
Proof :
We have, I = \(\int\) \(sec^{-1}x\) dx
Let \(sec^{-1}x\) = t,
Then, x = sec t
\(\implies\) dx = d(sec t) = sec t tan t dt
\(\therefore\) I = \(\int\) \(sec^{-1}x\) dx
\(\implies\) I = \(\int\) t (sec t tan t) dt
By using integration by parts formula,
I = t sec t – \(\int\) 1. (sec t) dt
I = t sec t – log |sec t + tan t| + C
Since tan t = \(\sqrt{sec^2 t – 1}\)
\(\implies\) I = t sec t – \(log |sec t + \sqrt{sec^2t – 1}|\) + C
Now, Put t = \(sec^{-1}x\) here,
\(\implies\) I = x \(sec^{-1}x\) – \(log|x + \sqrt{x^2 – 1}|\) + C
Hence, \(\int\) \(sec^{-1}x\) dx = x \(sec^{-1}x\) – \(log|x + \sqrt{x^2 – 1}|\) + C
Integration of Cosec Inverse x
The integration of cosec inverse x or arccosec x is x\(cosec^{-1}x\) – \(log|x – \sqrt{x^2 – 1}|\) + C
Where C is the integration constant.
i.e. \(\int\) \(cosec^{-1}x\) = x\(cosec^{-1}x\) – \(log|x – \sqrt{x^2 – 1}|\) + C
Proof :
We have, I = \(\int\) \(cosec^{-1}x\) dx
Let \(cosec^{-1}x\) = t,
Then, x = cosec t
\(\implies\) dx = d(cosec t) = -cosec t cot t dt
\(\therefore\) I = \(\int\) \(cosec^{-1}x\) dx
\(\implies\) I = \(\int\) t (-cosec t cot t) dt
By using integration by parts formula,
I = t cosec t – \(\int\) 1. (cosec t) dt
I = t cosec t – log |cosec – cot t| + C
Since cot t = \(\sqrt{cosec^2 t – 1}\)
\(\implies\) I = t cosec t – \(log |cosec t – \sqrt{cosec^2t – 1}|\) + C
Now, Put t = \(cosec^{-1}x\) here,
\(\implies\) I = x \(cosec^{-1}x\) – \(log|x – \sqrt{x^2 – 1}|\) + C
Hence, \(\int\) \(cosec^{-1}x\) dx = x \(cosec^{-1}x\) – \(log|x – \sqrt{x^2 – 1}|\) + C
Related Questions
What is the Differentiation of cosec inverse x ?
