Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have just started looking at integration and I am having trouble understanding what has been done in one of the examples in the book I am working through.

It involves using the double angle formula for $\sin(2\theta)$ to provide a rearrangement for which an indefinite integral can then be found.

The double angle formula provided is $\sin(2\theta)=2\sin(\theta)\cos(\theta)$ and the example is as follows:

$$\int\cos\left(\frac{1}{2}x\right)\sin\left(\frac{1}{2}x\right)dx=\int\frac{1}{2}\sin\left(x\right)dx$$ $$=-\frac{1}{2}\cos\left(x\right)+c$$

The part of this example I am specifically stuck with is the first line where $\cos\left(\frac{1}{2}x\right)\sin\left(\frac{1}{2}x\right)$ is rewritten as $\frac{1}{2}\sin\left(x\right)$ using the previously stated double angle formula.

share|improve this question
1  
substitute $\theta = \frac{x}{2}$ –  pedja Feb 24 '12 at 13:12
add comment

2 Answers

up vote 1 down vote accepted

You know that $$\sin(2 \theta)=2\sin(\theta)\cdot\cos(\theta)$$ Now, put $\theta=\dfrac{1}{2}x$ to see that, $$\begin{align}\sin\left(2 \cdot\dfrac{1}{2}x\right)&=2\sin\left(\dfrac{1}{2}x\right)\cdot\cos\left(\dfrac{1}{2}x\right)\\\sin(x)&=2\sin\left(\dfrac{1}{2}x\right)\cdot\cos\left(\dfrac{1}{2}x\right)\\\sin\left(\dfrac{1}{2}x\right)\cdot\cos\left(\dfrac{1}{2}x\right)&=\dfrac{1}{2}\cdot\sin(x)\end{align}$$

share|improve this answer
    
Thanks, helped a lot. Simple when you know how! –  Aesir Feb 24 '12 at 13:24
add comment

Consider $\theta = \frac{1}{2}x$. Then $$ \cos\left(\frac{1}{2}x\right)\sin\left(\frac{1}{2}x\right) = \cos \theta \sin \theta$$ Notice that the double angle formula could be written: $$ \sin \theta \cos \theta = \frac{1}{2}\sin(2\theta)$$ So the integrand is now: $$ \cos\left(\frac{1}{2}x\right)\sin\left(\frac{1}{2}x\right) = \cos \theta \sin \theta = \frac{1}{2} \sin(2\theta) = \frac{1}{2}\sin\left(2\cdot \frac{1}{2}x\right) = \frac{1}{2}\sin x$$

Hope this helps!

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.