7-6-45 integral of inverse tangent
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We can see that those integrals of the form DU over a ^2 + U ^2
which equals 1 / a tangent inverse U / a + C So if we look
at this, this 144 U squared is really 12U quantity squared.
So if we thought about this, we'd have 144 DU over 11 ^2 + 12
U squared.
So when we look at this, that would end up giving us we can
pull the 144 out in front if we want.
So we'd get DU over 11 ^2 + 12 U squared.
So the derivative of this though is going to be 12.
So we're going to have 144.
And instead of multiplying, we're going to divide because
we're undoing the integral.
And then we're going to have times the 1 / a, so 1 / 11
tangent inverse 12U over 11 + C So this reduces to twelve 11th
tangent inverse 12U over 11 + C Now if we wanted to check it, we
could take the derivative.
And if we thought about taking the derivative of this, we'd get
the 12 elevenths.
The derivative of tangent inverse is going to be 1 / 1
plus that U ^2 times the derivative of U DX.
Well, the derivative of U actually ddu OF12U over 11, so
we'd get 12 elevenths times 1 + 140, four U ^2 / 121 and the
derivative of this one is going to be 12 elevenths.
So then if we simplify this up, we get 12 elevenths times 12
elevenths is 144 / 121, so 144 / 121 * 1 / 1 plus 144 / 121 U
squared.
If we distribute this through, we'd get 144 over 121 + 144 U
squared and that's what we expected to get.