# What will be the value of this? Integration secxd(sec x)

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Feb 9, 2018

${\sec}^{2} x - \frac{1}{2} {\tan}^{2} x + C$

#### Explanation:

This is integration of the product of two functions. One is sec x and the other is $\frac{d}{\mathrm{dx}} \left(\sec x\right)$

Let first function be secx and the second function be $\frac{d}{\mathrm{dx}} \left(\sec x\right)$.

Application of the product rule of integration is:

first function * integral of second - $\int$ differentiation of first *integral of second

Now applying this rule would result in following:

sec x * sec x - $\int$ secx tanx * sec x dx

=${\sec}^{2} x - \int {\sec}^{2} x \tan x \mathrm{dx}$ Hint: [let tan x =t so that sec^ x dx =dt]

=${\sec}^{2} x - \frac{1}{2} {\tan}^{2} x + C$

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