First slide

Methods of integration

Question

the value of $\int_{}^{} e^{\tan θ} (\sec θ - \sin θ) d θ$ is

Difficult

A

$e^{\tan θ} \sec θ + c$
B

$e^{\tan θ} \sin θ + c$
C

$e^{\tan θ} (\sec θ + \sin θ) + c$
D

$e^{\tan θ} \cos θ + c$

Solution

Let $I = \int_{}^{} e^{\tan θ} (\sec θ - \sin θ) d θ$

Put $\tan θ = t \Rightarrow \sec^{2} θ d θ = d t \Rightarrow d θ = \frac{d t}{1 + t^{2}}$

$\Rightarrow I = \int_{}^{} e^{t} (\sqrt{1 + t^{2}} - \frac{t}{\sqrt{1 + t^{2}}}) \frac{d t}{1 + t^{2}}$

$= \int_{}^{} e^{t} (\frac{1}{\sqrt{1 + t^{2}}} - \frac{1}{{(1 + t^{2})}^{3 / 2}}) d t$

$= \int_{}^{} e^{t} (f + f') d t = e^{t} . f (t)$

$= \frac{e^{t}}{\sqrt{1 + t^{2}}} + c = e^{\tan θ} \cos θ + c$

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