Solving the Integral of (4-x)√(4x-x²)

[Stevecgz]

$$\int (4-x)\sqrt{4x-x^2}dx$$

I'm uncertain how to find this integral, or even how to start it. Any guidence would be appreciated. Thanks.

Steve

 

[James R]

Hmm... integration by parts, perhaps?

 

[TD]

It would be a lot easier if the integral were

$$\int {\left( {4 - 2x} \right)\sqrt {4x - x^2 } dx}$$.

If not, you'll need integration by parts indeed and I suspect you'll be getting an arcsin too

Think about

$$\sqrt {4x - x^2 } = \sqrt {4 - \left( {x - 2} \right)^2 } = 2\sqrt {1 - \left( {\frac{{x - 2}}<br /> {2}} \right)^2 }$$

 

[Stevecgz]

Thanks TD, got it worked out now.

Steve

 

[TD]

Good