Integrating Trigonometric functions (part 4)

We shall now proceed to integrating $secx$ and similarly, lets refresh the formulas we should know.

$\frac {d}{dx}tanx = sec^{2}x$

$\frac {d}{dx}secx = secxtanx$

$\int tanx dx = ln|secx|+c$ (MF15)

$\int secx dx = ln|secx+tanx|+c$ (MF15)

$\int sec^{2}x dx = tanx+c$

$\int sec^{3}x dx = \int secx(sec^{2}x)dx = \int secx(tan^{2}x+1)dx = \int secxtan^{2}x+secx dx$
So how do we $\int secxtan^{2}x dx$ ? I’ll first rewrite it as $\int (secxtanx)(tanx)dx$ for some insights.

We can’t adopt the $\int f'(x)f(x) dx$ method here. So, Integration by parts?

$\int (secxtanx)(tanx)~dx$

$= secx(tanx) - \int secx(sec^{2})~ dx$

$= secxtanx-\int sec^{3}x~dx$

Wait! $\int sec^{3}x dx$ again? hmmm.

So we have that

$\int sec^{3}x ~dx$

$= \int secxtan^{2}x+secx ~dx$

$= secxtanx-\int sec^{3}xdx + \int secx ~dx$ .

Then with a bit of juggling and manipulations, we have

$2\int sec^{3}x dx = secxtanx + ln|secx+tanx|+c$ .

I do hope this gives you some insights. You should try $\int sec^{4}x dx$ on your own using the information here.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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Integrating Trigonometric functions (part 4)

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Integrating Trigonometric functions (part 4)

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