Determine the integral from one to four of eight 𝑥 minus eight divided by the square root of 𝑥 minus one with respect to 𝑥 to the nearest hundredth.
The question is asking us to evaluate a definite integral to the nearest hundredth. And we can see this is the integral of the quotient of two functions. We can’t evaluate this integral directly. And at this point, we have a lot of different tools for dealing with integrals. For example, we might want to try using integration by parts or integration by substitution. And these are both great tools for dealing with integrals. However, the first question we should always ask is, can we rewrite our integrand into a form which we can integrate?
So let’s take a look at our integrand. It’s the algebraic expression, eight 𝑥 minus eight divided by the square root of 𝑥 minus one. We’ll start by taking out a factor of eight from our numerator. This gives us eight times 𝑥 minus one divided by root 𝑥 minus one. And at this point, we actually have several different algebraic tools which we can use to rewrite this expression. For example, we could multiply our numerator and our denominator by the conjugate of root 𝑥 minus one. The reason we would do this is it will give us a more simple expression for our denominator. Our denominator will become 𝑥 minus one. It’s also worth noting we could’ve used a difference between squares to rewrite 𝑥 minus one as root 𝑥 minus one times root 𝑥 plus one in our numerator.
And if we use either of these two methods, we’ll end up with the same answer. We’ll do this by multiplying the numerator and the denominator by root 𝑥 plus one. This gives us eight times 𝑥 minus one multiplied by root 𝑥 plus one all divided by 𝑥 minus one. We’ll then cancel our shared factor of 𝑥 minus one from the numerator and the denominator. And it’s worth noting we can do this because one is at the end of our interval of integration. This won’t change the area of our shape.
Therefore, if we distribute eight over our parentheses, we’ve rewritten our integrand as eight root 𝑥 plus eight. And this is an expression we definitely can integrate. So let’s use this to evaluate our integral. We’ve rewritten it as the integral from one to four of eight root 𝑥 plus eight with respect to 𝑥.
To do this, we’ll use our laws of exponents to rewrite root 𝑥 as 𝑥 to the power of one-half. We can now evaluate this integral term by term by using the power rule for integration. We want to add one to our exponent of 𝑥 and then divide by this new exponent of 𝑥. This gives us eight 𝑥 to the power of three over two divided by three over two plus eight 𝑥 evaluated at the limits of our integral one and four. The last thing we need to do is evaluate this at the limits of our integral.
Doing this, we get eight times four to the power of three over two divided by three over two plus eight times four minus eight times one to the power of three over two divided by three over two plus eight times one. And if we evaluate this expression, we get 184 divided by three. But remember, the question wanted this to the nearest hundredth, which is the same as two decimal places of accuracy. This gives us 61.33. Therefore, we’ve shown the integral from one to four of eight 𝑥 minus eight divided by root 𝑥 minus one with respect to 𝑥 to the nearest hundredth is 61.33.