If you have trouble with trigonometric substitutions, perhaps you will find this roundabout way more convenient:
<br />
I = \int{\sqrt{9 - x^{2}} \, dx} = \int{\frac{9 - x^{2}}{\sqrt{9 - x^{2}}} \, dx} = 9 \, \int{\frac{dx}{\sqrt{9 - x^{2}}}} - \int{\frac{x^{2}}{\sqrt{9 - x^{2}}} \, dx}<br />
In the first of these integrals, you need to perform a simple substitution:
<br />
x = a \, t<br />
<br />
dx = a \, dt<br />
where we will chose the constant a suitably so that the resulting integral simplifies:
<br />
9 \, \int{\frac{a \, dt}{\sqrt{9 - (a t)^{2}}}} = 9 \, a \, \int{\frac{dt}{\sqrt{9 - a^{2} \, t^{2}}}}<br />
If we choose:
<br />
9 = a^{2} \Rightarrow a = 3<br />
we can take out the 9 as a common multiple in front of both terms from the expression under the square root. We have:
<br />
9 \cdot 3 \, \int{\frac{dt}{\sqrt{9(1 - t^{2})}}} = \frac{9 \cdot 3}{3} \, \int{\frac{dt}{\sqrt{1 - t^{2}}}} = 9 \, \int{\frac{dt}{\sqrt{1 - t^{2}}}}<br />
I claim that the obtained integral is a standard one and can be read from a table.
As for the second integral, one needs to perform integration by parts:
<br />
\int{\frac{x^{2}}{\sqrt{9 - x^{2}}} \, dx} = \int{x \, \frac{x \, dx}{\sqrt{9 - x^{2}}}}<br />
Take
<br />
u = x \Rightarrow du = dx<br />
<br />
dv = \frac{x \, dx}{\sqrt{9 - x^{2}}}<br />
To find v, we integrate:
<br />
v = \int{\frac{x \, dx}{\sqrt{9 - x^{2}}}}<br />
This integral can be solved by a very clever substitution:
<br />
s = \sqrt{9 - x^{2}}<br />
<br />
ds = \frac{1}{2 \, \sqrt{9 - x^{2}}} \, (-2 x) \, dx = -\frac{x \, dx}{\sqrt{9 - x^{2}}}<br />
so:
<br />
v = -\int{ds} = -s = -\sqrt{9 - x^{2}}<br />
where, in the last step, I substituted back the old variable x instead of s (a necessary step in the method of substitution). So, integration by parts gives:
<br />
\int{\frac{x^{2}}{\sqrt{9 - x^{2}}} \, dx} = u \, v - \int{v \, du} = -x \, \sqrt{9 - x^{2}} + \int{\sqrt{9 - x^{2}} \, dx}<br />
Notice that the remaining integral is equal to the integral we had started with and this is the beauty of this approach. Now, we can combine everything to have:
<br />
I = 9 \, \int{\frac{dt}{\sqrt{1 - t^{2}}}} + x \, \sqrt{9 - x^{2}} - I<br />
Notice the different sign in front of I on both sides of the equality. That is why they don't cancel.
To finish the solution, you need to:
1. Express the remaining integral from a table of standard integrals. Do not forget to go back from t to x by the substitution we used in the beginning;
2. Solve this equation for I (by moving all I on one side of the equation and dividing by the coefficient in front of it);
3. Add an arbitrary integrating constant in the end.
IF you can't follow the above steps, either you have not learned about the methods of integration using substitutions and integration by parts or, in case you didn't follow the intermediate steps, your knowledge in algebra and differentiation is so poor that you actually need to learn those things before you can go on to integration techniques.
