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JH
Numerade Educator

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Problem 47 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{1 - \tan^2 x}{\sec^2 x} dx $

Answer

$$
\int \frac{1-\tan ^{2} x}{\sec ^{2} x} d x=\frac{1}{2} \sin 2 x+c
$$

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Video Transcript

this problem is from Chapter seven, Section two. Problem forty seven In the book Calculus Early Transcendental lt's a Condition by James Door. We have an indefinite integral of one minus tan squared on the numerator divided by C can swear. So let's go to the right and make some observations. One of our protagonist identities tells us that one plus tan square to see can't square. So this means that we have tan squared, seeking squared minus one, and this means one minus hand square is one minus seek inns where minus one, which is to minus seek and square. What so we can replace under Renner with two minus, he can't square. And the interval we have seeking square still in the denominator. Let God, it's split us up into two fractions. We have to oversee cans where should be into up there and then we have seeking squared over seek and squared, which is just one. Then we could simplify this even further. This is two coasts and squared eggs minus one D X, and this can be integrated using a formula for co sign squared. But actually, in this case, you can go ahead and rewrite then a grand using the double angle formula. This is simply co sign of two works. If you didn't realize this fact than another way to proceed would have been to take this coastline square and right It is one plus coastline of two X over two. Even after doing this, you will still ended up at the same equation that we're right now. Then we know the integral this and my help here excellent might help you to do a U substitution. After doing so, we should get sign of two X over to plus he and that's our answer.