MHB Mohammad nabeel's question at Yahoo Answers regarding a definite integral

AI Thread Summary
The integral of sin^3(6x)cos^4(3x) from 0 to 60 degrees can be evaluated using a change of variables. By substituting u = x - π/6, the integrand transforms into an odd function. The limits of integration also adjust accordingly, leading to an integral that evaluates to zero due to the properties of odd functions. Thus, the final result of the integral is zero. This demonstrates a useful technique for simplifying complex integrals.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the original question:

Ntegral of sin^3(6x)cos^4(3x) dx ? between limits o to 60 degrees

Here is a link to the question:

Ntegral of sin^3(6x)cos^4(3x) dx ? - Yahoo! Answers

I have posted a link there to this topic so that the OP can find my response.
 
Mathematics news on Phys.org
Hello Mohammad,

We are given to calculate:

$\displaystyle \int_0^{\frac{\pi}{3}}\sin^3(6x)\cos^4(3x)\,dx$

If we use a change of variables, i.e.,

$\displaystyle u=x-\frac{\pi}{6}\,\therefore\,du=dx\,\therefore\,x=u+\frac{\pi}{6}$

then we may rewrite the integrand as follows:

$\displaystyle \sin(6x)=\sin\left(6\left(u+\frac{\pi}{6} \right) \right)=\sin(6u+\pi)=\sin(6u)\cos(\pi)+\cos(6u) \sin(\pi)=-\sin(6u)$

$\displaystyle \cos(3x)=\cos\left(3\left(u+\frac{\pi}{6} \right) \right)=\cos\left(3u+\frac{\pi}{2} \right)=\cos(3u)\cos\left(\frac{\pi}{2} \right)-\sin(3u)\sin\left(\frac{\pi}{2} \right)=-\sin(3u)$

and our integral becomes (don't forget to change the limits in accordance with the change of variable):

$\displaystyle -\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\sin^3(6u)\sin^4(3u)\,dx$

Now we have an odd-function as the integrand, and by the odd function rule, this is simply zero.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top