Question

Prove the integration formula $\int secxdx=\ln |secx+\tan x|+C$

(a) by multiplying the integrand by a form of 1 and then applying a substitution;

(b) by differentiating ln | sec x + tan x|.

Short answer

(a) Formula is proved by multiplying the integrand by a form of 1

(b) Formula is proved by differentiating ln | sec x + tan x|.

Step-by-step solution

Part (a) Step 1. Explanation

$\begin{array}{rcl}\int secxdx& =& \int \frac{secx\left(secx+\tan x\right)}{secx+\tan x}dx\\ & =& \int \frac{se{c}^{2}x+secx\tan x}{secx+\tan x}\end{array}$

Let width="117">secx+tanx=u

$secx\tan x+se{c}^{2}xdx=du$

$\begin{array}{rcl}\int secxdx& =& \int \frac{du}{u}\\ & =& \ln u+C\\ & =& \ln \left(secx+\tan x\right)+C\end{array}$

Part (b) Step 1. Explanation

$\begin{array}{rcl}\frac{d}{dx}\left(\ln \left|secx+\tan x\right|\right)& =& \frac{secx\tan x+se{c}^{2}x}{\left|secx+\tan x\right|}\\ & =& \frac{secx\left(\tan x+secx\right)}{secx+\tan x}\\ & =& secx\\ & \Rightarrow & \int secx=\ln \left|secx+\tan x\right|+C\\ & & \end{array}$