Integrating Trigonometric functions (part 3)

So part 3, we look at $tanx$ ! This is slightly more complicated as we need to first lay out some formulas that 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)

Notice how $tanx$ and $secx$ are quite related here… And we recall we have a trigonometry identity that happens to be that closely related:

$tan^{2}x+1=sec^{2}x$ !

$\int tan^{2}x dx = \int sec^{2}x -1 ~dx = tanx - x + c$

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

$= \int tanx(tan^{2}x) ~dx$

$= \int tanx(sec^{2}x-1) ~dx$

$= \int tanxsec^{2}x-tanx ~dx$

Here we have a slight problem with $\int tanxsec^{2}x$

Recall the $\int f'(x)f(x)$ method we saw in part 1 and 2…
$\int tanxsec^{2}x= \frac {tan^{2}x}{2}+c$

So, $\int tanxsec^{2}x-tanx ~dx=\frac {tan^{2}x}{2}-ln|secx|+c$

$\int tan^{4}x dx = \int tan^{2}x(sec^{2}x-1) ~dx = \int tan^{2}xsec^{2}x-tan^{2}x dx$

As you probably guessed, $\int tan^{2}xsec^{2}x ~dx = \frac {tan^{3}x}{3}+c$ and for $\int tan^{2}x dx$ , we can refer to the previous results.

We will look at $secx$ in part 4, and surprise surprise, it will be quite a different appraoch! Let me know if there are any questions.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for almost two decades. 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 3)

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Integrating Trigonometric functions (part 3)

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