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Problem 45 Hard Difficulty

Evaluate the indefinite integral.

$ \displaystyle \int \frac{1 + x}{1 + x^2} \, dx $

Answer

$$\tan ^{-1} x+\frac{1}{2} \ln \left(1+x^{2}\right)+C$$

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Video Transcript

okay for this problem of first thing you must recall is that we can break up this integral in tow one over one plus X Square detox, plus the integral of acts over one plus x squared DX. And the reason why is because one plus tax everyone pulls X squared DX is equivalent to these two. We can break them up. Given the fundamental dilemma calculus. Now we know that you is one plus x squared. Take the derivative. We now have inverse tangent of axe post 1/2 times do you over you, which gives us inverse tangent of X. That's the first part plus 1/2 and now we can substitute natural log of one plus x squared. Remember, this is what our US plus C