Ιntegral calculation : (sin(x))^4 * (cos(x))^6

Summary: Ιntegral calculation : (sin(x))^4 * (cos(x))^6

Hi all,
I tried to solve it, but I got stuck. An advice from my professor is to set: x=arctan(t)

Τhanks.

PeroK
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Alternatively, you can get a formulas for the integral of ##\sin^n x## and ##\cos^n x## from repeated integration by parts.

They are on my list of standard integrals.

fresh_42
Mentor
Since both trigonometric functions lead to one another and back by differentiation and integration, integration by parts is usually how they are solved. Doing it twice should reduce the power of one while keeping the other one stable. This leads to a recursion, or can just be repeated until a single function is left.

I have found the formula on Wikipedia.

Michael_0039
Thanks for your answers! I tried another way using the following trigonometric transformation formulas:

Now it's in simple format:

Do you know a better - faster way to solve it ?

Thanks.

fresh_42
Mentor
Yes, integration by parts. Still.
$$\int \sin^n x \cos^m x \,dx = -\dfrac{\sin^{n-1}x \cos^{m+1}x}{n+m} +\dfrac{n-1}{n+m}\int \sin^{n-2} x \cos^m x \,dx$$

PeroK and Michael_0039
PeroK
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2020 Award
Do you know a better - faster way to solve it ?

Thanks.
I would have started with ##\sin^4 x = (1 - \cos^2 x)^2 = 1 - 2\cos^2x + cos^4x##.

Then used parts to calculate the integral of the various powers of ##\cos x##.

This is so useful and common that I keep the recursion formula in my list of standard integrals.

fresh_42 and Michael_0039
PeroK
$$\int \sin^n x \cos^m x \,dx = -\dfrac{\sin^{n-1}x \cos^{m+1}x}{n+m} +\dfrac{n-1}{n+m}\int \sin^{n-2} x \cos^m x \,dx$$