7-6-61 integral of inverse tangent
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We're going to let U equal sine 5 Theta so that U of negative π
tents is going to equal sine of 5 * -π tents or sine of negative
π halves.
Sine of negative π halves is -1.
We also need U of π tenths, so sine of 5 * π tenths which would
be sine of π halves which would be one DU is going to be 5
cosine 5 Theta D Theta using the chain rule.
So now we're going to end up with lower bound -1 upper bound
15 cosine 5 Theta D Theta.
Well, we have a 25 cosine 5 Theta D Theta.
So if I thought about multiplying each side by 5,
there is lots of ways to do this, but this is one of them.
If I multiply each side by 5I end up with that 25 cosine 5
Theta D Theta.
So I'm going to pull out the five in front.
I'm going to have DU over 1 + U ^2.
Now, this is just one of our formulas.
This is really just the integral of DU of a ^2 + U ^2.
So the five is going to be there.
The formula says it's going to be 1 / A.
Well, this is our a ^2 right here.
So A in this case is just one tangent inverse of U oops U / a,
and that's going to be bounded.
So we're going to have five tangent inverse of 1 -, 5
tangent inverse of -1.
Well, with our bound restriction for tangent, we're restricted
between π halves and negative π halves because we want it to be
a smooth continuous function if possible.
So the tangent inverse of one is going to be π force.
The tangent inverse of -1 is going to be negative π force.
So we get 5 Pi force, +5 Pi force, five Pi force +5 Pi
force, 10 Pi force, which reduces to five Pi halves.