First slide

Methods of integration

Question

$\int e^{- x} (1 - \tan x) \sec x d x is equal to$

Moderate

A

$e^{- x} \sec x + c$
B

$e^{- x} \tan x + c$
C

$- e^{- x} \tan x + c$
D

$- e^{- x} \sec x + c$

Solution

$Putting x = - z, we get$
$\begin{array}{l} I = \int - e^{z} (\sec z + \sec z \cdot \tan z) dz \\ = - \int e^{z} \{\sec z + \frac{d}{dz} (\sec z)\} dz = - e^{z} \sec z + k \\ ∴ I = - e^{- x} \sec x + k \end{array}$

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