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Problem 49 Hard Difficulty

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int x(x^2 - 1)^3 \, dx $

Answer

$$\frac{1}{8}\left(x^{2}-1\right)^{4}+C$$

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Video Transcript

we are. We know we're trying to figure out the integral of axe times X squared minus one. Q d ox. The first thing we know is that if RU is ex scored minus one dinar d, you must be to axe de acts. The derivative of negative one is simply zero, which means we have this. We're pulling out 1/2 because it's the constant, which means using our power rule, increasingly exploited by one dividing By Then you explain it, we end up with us. Graphing this we know the anti derivative a k a. The integral looks like this and the function it looks like thus you can clearly see the extreme appoints. You could see the maximum minimum Maximus, where we see changing signs from positive to negative plus to minus. And minima is work shooting signs from negative to positive. Therefore, the answer is reasonable