And so this is going to beįive to the fourth over four minus two to the fourth over four. We're gonna evaluate that at five and two. Let's see this is going to be equal to, the antiderivative of u to the third power is u to the fourth over four.
SOLVE USING SUBSTITUTION PLUS
When x is equal to two, two squared plus one, u is equal to five. We used our substitution right over here. We are now integrating with respect to u, and the way we did it is But it's really important to realize why we changed our bounds. This as being equal to the integral from two to five of u to the third du. With respect to u, you assume it's u equals It's often just from two to five because we're integrating See someone writing u equal two or u equals five. When x is equal to two, what is u? Well you have two squared which is four plus one, which is five, so u is equal to five. And so your bounds of integration, let's see when x isĮqual to one, what is u? Well when x is equal to one, you have one squared plus one, so you have two, u is equal This a definite integral I guess you could say, you would change your boundsįrom u is equal to something to u is equal to something else. Your bounds of integration, or if you wanna keep Because this one is xĮquals one to x equals two. You can change your bounds of integration. So what happens to ourīounds of integration? Well there's two ways that Now an interesting question, because this isn't an indefinite integral, we're not just trying toįind the antiderivative. So two x times dx, well two x times dx, that is du. Remember you could just view this as two x times x squared plus one to So we have u to the third power, u, the same orange color, u to the third power. So let me at least write, so this is going to be, I'll write the integral. And so at least this part of the integral I can rewrite. Way of thinking about it is multiplying both sides by dx. Of u with respect to x is just two x plus zero or just two x. I could say, u is equal to x squared plus one, in which case the derivative Squared plus one business to the third power, but then I also have theĭerivative of x squared plus one which is two x right over here. And the key giveaway here is, well I have this x Gonna apply u-substitution, but it's interesting to be able
So let's say we have the integral, so we're gonna go from xĮquals one to x equals two, and the integral is two x times x squared plus one to the third power dx.
Going to do in this video is get some practice applying u-substitution
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