Evaluate the indefinite integral:∫sqrt(10x−x^2)dx ?
2 Answers
Explanation:
Complete the square,
Substitute
Substitute
Simplify,
Refine,
Take out the constant,
Apply double angle formulae,
Take out the constant,
Integrate,
Substitute back
Simplify,
Refine,
Tadaa :D
Explanation:
What is
Note that the domain of the function being integrated is where the inner quadratic is positive, i.e.
This expression can be integrated using substitutions. Though a possible pathway for integration doesn't immediately present itself, if we compete the square, then a trigonometric substitution can be carried out:
Which, we notice, is in the classic trigonometric substitution form, i.e. the square of a number minus the square of a linear
First, to get rid of the linear, we let
Now for the second substitution, let
Of course, the
Now we can use a double angle formula to make integrating
So the integral becomes:
Now,
Hence,
And,