# Thread: Integration Using Double Angle Formulas - Ex 3

1. ## Integration Using Double Angle Formulas - Ex 3

$\displaystyle \int \tan 5x dx$

How can a double angle formula be used in this situation? hint?

2. ## Re: Integration Using Double Angle Formulas - Ex 3

I would use u-sub u= 5x and write $\displaystyle tan(u)$ as $\displaystyle \frac{sin(u)}{cos(u)}$

the double angle would be $\displaystyle \displaystyle \int \frac{2tan(\frac{5}{2}x)}{1-tan^2(\frac{5}{2}x)}dx$ I'm not interested in solving it this way.

3. ## Re: Integration Using Double Angle Formulas - Ex 3

One more time: u= 5x gives a very simple integral.

(If your problem had been $\displaystyle \int sin^2(x) dx$, $\displaystyle \int cos^4(x) dx$, and $\displaystyle \int tan^5(x) dx$ that would be a different matter.)

4. ## Re: Integration Using Double Angle Formulas - Ex 3

How about? This is what would need double angle stuff:

$\displaystyle \int tan^{2}(5x) dx$

5. ## Re: Integration Using Double Angle Formulas - Ex 3

This is easiest done using a Pythagorean Identity:

\displaystyle \begin{align*} \int{ \tan^2{(5x)}\,\mathrm{d}x} &= \int{ \sec^2{(5x)} - 1 \, \mathrm{d}x} \end{align*}

You should be able to go from here...

6. ## Re: Integration Using Double Angle Formulas - Ex 3

K double-angle rule should be reserved for cos^2, sin^2 and cos^(2n)*sin^(2m) only.
Tangent and secant? No, please don't use double-angle on them.

7. ## Re: Integration Using Double Angle Formulas - Ex 3

Originally Posted by Prove It
This is easiest done using a Pythagorean Identity:

\displaystyle \begin{align*} \int{ \tan^2{(5x)}\,\mathrm{d}x} &= \int{ \sec^2{(5x)} - 1 \, \mathrm{d}x} \end{align*}

You should be able to go from here...
$\displaystyle \int tan^{2}(5x) dx$

$\displaystyle \int tan^{2}(u)dx$

$\displaystyle \int sec^{2}(u) - 1 dx$

$\displaystyle \dfrac{1}{5}\dfrac{\tan^{3} 5x}{3} - x + C$

$\displaystyle \dfrac{\tan^{3} 5x}{15} - x + C$

$\displaystyle \dfrac{\tan^{3} x}{3} - x + C$

8. ## Re: Integration Using Double Angle Formulas - Ex 3

No. What is the derivative of tan(x) ?

9. ## Re: Integration Using Double Angle Formulas - Ex 3

Originally Posted by Jason76
$\displaystyle \int tan^{2}(5x) dx$

$\displaystyle \int tan^{2}(u)dx$

$\displaystyle \int sec^{2}(u) - 1 dx$
You really need to write these with the correct variable. The first integration is over x. Good. The remaining two are over u. So you need to use dx = (1/5) du.

-Dan