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Evaluate the definite integral from three to five of ๐ฅ multiplied by the square root of five ๐ฅ squared plus three with respect to ๐ฅ to the nearest thousandth.
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Letโs look at the function weโve been asked to integrate more closely.
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It consists of a polynomial function, ๐ฅ, multiplied by the square root of another polynomial function, five ๐ฅ squared plus three.
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We have a variety of integration methods available to us, methods such as integration by substitution or integration by parts.
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And in order to decide which method is appropriate, we need to consider what form the integrand is in.
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Well, if we look closely, we can see that the derivative with respect to ๐ฅ of the function under the square root, thatโs five ๐ฅ squared plus three, is 10๐ฅ, which is a scalar multiple of the function weโre multiplying by, ๐ฅ.
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And so our integrand is of the form ๐ of ๐ of ๐ฅ, a composite function, multiplied by ๐ prime of ๐ฅ, the derivative of the inner function, subject to multiplication by a constant.
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This tells us that we could answer this question using the method of integration by substitution.
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To do this, we introduce a new variable.
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So weโre going to let ๐ข equal five ๐ฅ squared plus three.
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Thatโs the inner function in our composite function.
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And we want to change everything in this integral from something in terms of ๐ฅ to something in terms of ๐ข.
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Weโve already said that the derivative of five ๐ฅ squared plus three, so thatโs now d๐ข by d๐ฅ, is equal to 10๐ฅ.
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Now, d๐ข by d๐ฅ is absolutely not a fraction.
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But we can treat it a little like one when weโre performing this method.
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So itโs equivalent to say that d๐ข is equal to 10๐ฅ d๐ฅ.
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All we can say that one-tenth of d๐ข is equal to ๐ฅ d๐ฅ.
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And the reason this is useful is because in our original integral, we have ๐ฅ d๐ฅ, which we can now replace with one-tenth d๐ข.
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So, letโs now make this substitution and change our integral to be in terms of ๐ข.
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Firstly, the square root of five ๐ฅ squared plus three can be written as the square root of ๐ข.
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Then, ๐ฅd๐ฅ can be replaced with one-tenth d๐ข.
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But weโre not done.
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Thereโs one more important step.
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Because this is a definite integral, we must remember to change its limits from limits in terms of ๐ฅ to limits in terms of ๐ข.
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For our upper limit, we substitute ๐ฅ equals five into the equation connecting ๐ฅ and ๐ข.
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And we get ๐ข equals five multiplied by five squared plus three, which is 128.
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For our lower limit, we substitute ๐ฅ equals three into the same equation, giving ๐ข equals five multiplied by three squared plus three, which is 48.
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Our integral has now become the definite integral from 48 to 128 of the square root of ๐ข one-tenth d๐ข.
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And so weโve changed it to be fully in terms of ๐ข.
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We can rewrite the square root of ๐ข as ๐ข to the power of one-half.
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And if it helps, we can also bring that factor of one-tenth out the front of the integral.
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So we have one-tenth, the integral from 48 to 128, of ๐ข to the power of one-half with respect to ๐ข.
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To integrate a power of ๐ข, where the power is not equal to negative one, we recall that we increase the power or the exponent by one.
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So that gives ๐ข to the power of three over two.
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And then, we divide by the new exponent.
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Dividing by three over two is equivalent to multiplying by two over three.
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So our integral is equal to one-tenth of two-thirds ๐ข to the power of three over two evaluated between 48 and 128.
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And because we changed the limits in this definite integral, thereโs no need to reverse our substitution and go back to an answer in terms of ๐ฅ.
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Our constants of one-tenth and two-thirds can be combined to give an overall constant of one fifteenth.
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And we then substitute our limits, giving one fifteenth of 128 to the power of three over two minus 48 to the power of three over two.
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We can use a calculator to evaluate this.
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And it gives 74.37339 continuing.
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Finally, we recall that the question asks us to give our answer to the nearest thousandth.
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So rounding this value, we obtain 74.373.
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Using the method of integration by substitution then, we found that the definite integral from three to five of ๐ฅ multiplied by the square root of five ๐ฅ squared plus three with respect to ๐ฅ is equal to 74.373 to the nearest thousandth.