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Integrate log

in this video we're going to talk about how to integrate functions that turn into natural log functions so let's start with this example what is the antiderivative of 1 over X DX it turns out that this is equal to the natural log of X plus C the derivative of Ln X is 1 over X so the antiderivative of 1 over ax is Ln X now we need to put an absolute value around the X you can't have a negative number inside a natural log function so we need the absolute value now let's try another problem what is the antiderivative of 7 over X DX what would you say well we can move the constant to the front and we know that the antiderivative of 1 over X DX is Ln X so the answer is going to be 7 Ln X plus C and so that's all you need to do for this example here's another example what about the antiderivative of 1 over X plus 5 DX what do you think we need to do for this problem well this is equal to the natural log of X plus 5 plus C now you can prove it using new substitution so let's go ahead and do that we're going to say U is equal to X plus 5 and so therefore D U will be equal to DX so in this problem we can replace X plus 5 with the u variable and we can replace DX with D u if the antiderivative of 1 over X is Ln X then the antiderivative of 1 over U is Ln U plus C now all we need to do is replace the u variable with X plus 5 so the answer is going to be Ln X plus 5 plus C and let's use an absolute value symbol now how about this one the antiderivative of 3 over X minus 4 DX what is this equal to it turns out this is going to be 3 times the natural log of X minus 4 plus C so 1 over X minus 4 is Ln X minus 4 and because you have the 3 on top simply multiply that answer by 3 and you can use u-substitution improve it here's another one every to try what's the antiderivative of 8 over 5 minus Q X DX so we're going to have a constant 8 times the natural log of the stuff in the bottom five minus two x plus C now the derivative of the inside the derivative of 5 minus 2x is negative 2 so we have to divide it by negative 2 so the answer is going to be 8 over negative 2 which is negative 4 Ln 5 minus 2x plus C now feel free to pause the video and use u-substitution to prove that this is the answer go ahead and try it so we're going to make you equal to 5...