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

In summary: Therefore, our integral evaluates to:$\displaystyle \int_0^{\frac{\pi}{3}}\sin^3(6x)\cos^4(3x)\,dx=0$In summary, the integral of sin^3(6x)cos^4(3x) dx between the limits of 0 to 60 degrees is equal to 0. This is because the integrand can be rewritten using a change of variables and simplifies to an odd function, which evaluates to 0.
  • #1
MarkFL
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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.
 
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  • #2
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.
 

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

What is a definite integral?

A definite integral is a mathematical concept used to find the area under a curve between two points. It is represented by the symbol ∫ and is calculated using limits and an integrand function.

What does Mohammad nabeel's question at Yahoo Answers regarding a definite integral mean?

Without the specific question, it is difficult to determine the exact meaning of Mohammad nabeel's question. However, it is likely that they are seeking help or clarification on a specific definite integral problem they are working on.

How is a definite integral different from an indefinite integral?

A definite integral has specific limits of integration and gives a numerical value, while an indefinite integral does not have limits and gives a function as a result. In other words, a definite integral calculates the area under a curve, while an indefinite integral finds the antiderivative of a function.

What is the purpose of using definite integrals in science?

Definite integrals are used in science to calculate various physical quantities such as displacement, velocity, acceleration, and work. They also have applications in areas such as physics, engineering, and economics.

What is the process for solving a definite integral?

To solve a definite integral, you must first identify the limits of integration and the integrand function. Then, you can use various integration techniques, such as substitution or integration by parts, to evaluate the integral. Finally, you must plug in the limits and solve for the numerical value of the integral.

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