Video: Finding the Integration of a Function Involving Expanding a Square and Using the Power Rule

Determine ∫(3𝑥 − (7/𝑥))² d𝑥.

02:39

Video Transcript

Determine the integral of three 𝑥 minus seven over 𝑥 all squared d𝑥.

So the first thing we need to do if we want to solve this problem is rewrite what’s inside the parentheses and then distribute across our parentheses. So the first thing we’ve done to help us with that is use one of our exponent rules. And that is that if 𝑥 is to the power of negative one, then this is equal to one over 𝑥. So therefore, what we’ve got is three 𝑥 minus seven 𝑥 to the power of negative one multiplied by three 𝑥 minus seven 𝑥 to the power of negative one.

So now, if we distribute across the parentheses, we’re gonna have nine 𝑥 squared. That’s three 𝑥 multiplied by three 𝑥. Then, we’ve got minus 21. And that’s because three 𝑥 multiplied by seven 𝑥 to the power of negative one is gonna give us 21𝑥 and then 𝑥 to the power of one multiplied by 𝑥 to the power of negative one. This is gonna be 𝑥 to the power of zero, which is just one. So that leaves us with our 21.

And then, we’re gonna have minus 21 again. And then finally, plus 49𝑥 to the power of negative two. And that’s because negative seven 𝑥 to the power of negative one multiplied by negative seven 𝑥 to the power of one. Negative seven multiplied by negative seven gives us positive 49. Then, 𝑥 to the power of negative one multiplied by 𝑥 to the power of negative one gives us 𝑥 to the power of negative two. So then, if we simplify, we’ve got nine 𝑥 squared minus 42 plus 49𝑥 to the power of negative two. Okay, great. So we’ve done this. So now, what we’re gonna do is get on and integrate our expression.

So now, if we’re gonna integrate our expression, what we can do is integrate each term individually. So then, our first term is gonna be nine 𝑥 cubed over three. If we remind ourselves how we do that, how we integrate, what we do is we add one to the exponent. So it’s gonna be nine 𝑥 to the power of two plus one. And then, we divide by that same new exponent, which in this case is going to be two plus one, which is three. Then, we’re gonna have minus 42𝑥, then minus 49𝑥 to the power of negative one, then plus 𝑐, which is our constant of integration.

We’ve got the 49𝑥 to the power of negative one, because if you add one to negative two, you get negative one. If you divide by negative one, it means that we get negative 49. So we’ve got negative 49𝑥 to the power of negative one. So we’ve almost finished. We just need to simplify. So this is gonna give us our final answer of three 𝑥 cubed minus 42𝑥 minus 49 over 𝑥 plus 𝑐. And we can say this is the result of the integral of three 𝑥 minus seven over 𝑥 all squared d𝑥.

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