00:01
In this question, we're asked to calculate the given integral.
00:04
And to do that, well, we will use trigonometric substitution.
00:09
And in our case, that's going to be x equals to 1 or 9, multiplied by sine theta.
00:17
Then, d x equals to 1 or 9 cause theta, and therefore we can rewrite the integral in terms of theta, as the integral of the square root of 1 minus 81 multiplied by 1 over 9 times sine theta squared times 1 over 9 times cos theta d theta.
00:52
The next step is to simplify the expression inside the square root.
00:58
We are going to get the square root of 1 minus 81 times 1 over 81 times sine squared theta, multiplied by 1 over 9 cos theta d theta.
01:13
Then we can cancel ad1 and we are going to get the integral of the square root of 1 minus sine squared theta times 1 over 9 cause theta d theta.
01:31
Now the reason we did that substitution, the reason we chose x to be 1 over 9 sine theta, it was to cancel ad1.
01:41
Rate.
01:43
And after canceling 81, the expression under the square root is cosquired theta.
01:51
So, we are going to get the integral of the square root of cost squared theta times 1 over 9 cos theta d theta, and the square root of cosquered is just cosine.
02:04
So that's going to be 1 over 9 times the integral of cost squared theta d theta.
02:10
To calculate this integral, we will use the half angle formulas by the half angle formulas cost squared equals to one half multiplied by one plus plus cos two theta then we're going to get 1 or 18 times the integral of 1 equals to theta and the integral of cos 2 theta equals to 1 half times sine 2 theta and of course plus the constant of integration c now we need to get back to the original variable, right? recall that we made a substitution x equals to 1 over 9 sine theta.
02:57
X equals to 1 over 9 sine theta and that means that sine theta equals to 9x, right? now let's draw some right triangles...