$\displaystyle \int cos^4 2x sin^3 2x dx$
I've tried simplifying the equation first but got stucked instead, and tried substitution also.
Which method should i use for this?
Thanks in advance.
hmmm why don't you try to first factor sin^3(2x) to sin^2(2x)(sin(2x))... then since sin^2(2x) is also equal to 1-cos^2(2x).... you rearrange it to a indefinite integral of cos^4(2x)[1-cos^2(2x)]sin(2x)dx.. then do substitution... u=cos (2x).... du= (2)(-sin(2x))dx.... now... please do the rest
Try rewriting the integrand as follows:
$\displaystyle \cos^4(2x)\sin^3(2x)=\cos^4(2x)\sin^2(2x)\sin(2x)= \cos^4(2x)(1-\cos^2(2x))\sin(2x)=$
$\displaystyle \cos^4(2x)\sin(2x)-\cos^6(2x)\sin(2x)$
Can you proceed from here using an appropriate substitution?
At my university, students are taught general methods for the integrals $\displaystyle \int\cos^nx\sin^mx\ dx$ and $\displaystyle \int\sec^nx\tan^mx\ dx$. The substitution depends on whether n and m are even or odd - it's actually pretty straightforward if you look at the derivatives of those four trigonometric functions.
- Hollywood
Thanks guys question solved. Just another question
$\displaystyle \int cot^2 \pi x dx$
i tried changing it into
$\displaystyle \frac{1 + cos2\pi x}{1-cos2\pi x}$ and got stuck.
ps:can't seem to edit my original post.
I would use a Pythagorean identity to rewrite the integrand as:
$\displaystyle \csc^2(\pi x)-1$
and then use the fact that:
$\displaystyle \frac{d}{dx}(\cot(u(x)))=-\csc^2(u(x))\frac{du}{dx}$
to rewrite the integral as:
$\displaystyle -\frac{1}{\pi}\int\,d(\cot(\pi x))-\int\,dx$