# Thread: [SOLVED] Definite Integral, parts, with trig substution

1. ## [SOLVED] Definite Integral, parts, with trig substution

Okay I have: $\displaystyle \int \frac {sec^{14}{20x}}{cot(20x)}$
I know I should break up $\displaystyle sec{x}$ but I am not sure what to do for the trig substitutions.
Thanks,
Matt

2. Originally Posted by matt3D
Okay I have: $\displaystyle \int \frac {sec^{14}{20x}}{cot(20x)}$
I know I should break up $\displaystyle sec{x}$ but I am not sure what to do for the trig substitutions.
Thanks,
Matt
You're correct, you first need to split off a factor of $\displaystyle \sec(20x)$

$\displaystyle \int\frac{\sec^{14}(20x)}{\cot(20x)}\,dx=\int\frac {\sec^{13}(20x)\cdot\sec(20x)}{\cot(20x)}\,dx$

Now note that $\displaystyle \frac{1}{\cot(20x)}=\tan(20x)$

So we now see that $\displaystyle \int\frac{\sec^{13}(20x)\cdot\sec(20x)}{\cot(20x)} \,dx=\int \sec^{13}(20x)\cdot\sec(20x)\tan(20x)\,dx$

Now it should be clear what the proper substitution should be.

I hope this makes sense!

--Chris

3. Thanks Chris! I forgot to mention that it had limits from $\displaystyle 0$ to $\displaystyle pi/60$ after I did u substitution I got $\displaystyle \frac {16383}{280}$

4. Originally Posted by matt3D
Okay I have: $\displaystyle \int \frac {sec^{14}{20x}}{cot(20x)}$
I know I should break up $\displaystyle sec{x}$ but I am not sure what to do for the trig substitutions.
Thanks,
Matt
$\displaystyle \int_0^{\pi/60} \frac{(\sec(20x))^{14}}{\cot(20x)} \ dx= \int_0^{\pi/60} \frac{\sin(20x)}{(\cos(20x))^{15}} \ dx=$$\displaystyle \int_0^{\pi/60} \frac{1}{20 \times 14}\frac{d}{dx} (\cos(20x))^{-14} \ dx$

RonL