Question Video: Evaluating the Definite Integral of an Exponential Function with an Integer Base Mathematics • Higher Education

Evaluate โˆซ_(3)^(4) 4^(๐‘ ) d๐‘ .

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Video Transcript

Evaluate the definite integral from three to four of four to the power of ๐‘  with respect to ๐‘ .

Weโ€™re asked to evaluate the definite integral of an exponential function. Weโ€™ll do this by using the fundamental theorem of calculus. So weโ€™ll start by recalling the fundamental theorem of calculus. In fact, weโ€™ll only recall the part which relates to how we evaluate definite integrals. This says if lowercase ๐‘“ is continuous on the closed interval from ๐‘Ž to ๐‘ and capital ๐น prime of ๐‘ฅ is equal to lowercase ๐‘“ of ๐‘ฅ, then the definite integral from ๐‘Ž to ๐‘ of lowercase ๐‘“ of ๐‘ฅ with respect to ๐‘ฅ is equal to capital ๐น evaluated at ๐‘ minus capital ๐น evaluated at ๐‘Ž.

In other words, if our integrand is continuous across the entire interval of integration and we have an antiderivative for our integrand, then we can evaluate our definite integral by evaluating our antiderivative at the upper limit of integration and subtracting this from the antiderivative evaluated at the lower limit of integration. So letโ€™s work on this piece by piece. First, we need our integrand to be continuous on the interval of integration. By looking at our definite integral, we can see the lower limit of integration is three and the upper limit is four. So weโ€™ll set ๐‘Ž equal to three and ๐‘ equal to four. So we just need to show that our function lowercase ๐‘“ is continuous on the closed interval from three to four.

Thereโ€™s a few different ways we could do this. We could do this directly from the definition of four to the power of ๐‘ . However, itโ€™s easier to rewrite this in terms of the exponential function because we know a lot about the continuity of the exponential function. So weโ€™ll rewrite our integrand. Weโ€™ll do this by recalling that four is equal to ๐‘’ to the power of the natural logarithm of four. So by using this, we can rewrite our integrand as ๐‘’ to the power of the natural logarithm of four all raised to the power of ๐‘ . But we can then rewrite this expression by using our laws of exponents. We have that this is equal to ๐‘’ to the power of ๐‘  times the natural logarithm of four.

And now we can see this is just an exponential function, so we know itโ€™s continuous for all real values of ๐‘ . In particular, this means itโ€™s continuous on the closed interval from three to four. Now, to use the fundamental theorem of calculus, we need to find an antiderivative of our integrand. And once again, itโ€™s easier to use the form where we wrote this as an exponential function because we know a lot about finding antiderivatives of exponential functions. Now, we could try and find our antiderivative by using what we know about our derivative rules. However, in this case, because weโ€™ve written this as an exponential function, we can just use our rules of integrating exponential functions.

We know for any real constant ๐‘Ž not equal to zero, the integral of ๐‘’ to the power of ๐‘Ž๐‘  with respect to ๐‘  is equal to one over ๐‘Ž times ๐‘’ to the power of ๐‘Ž๐‘  plus a constant of integration ๐ถ. In our case, the value of the constant ๐‘Ž is the natural logarithm of four, which we know is not equal to zero. So we get one over the natural logarithm of four times ๐‘’ to the power of ๐‘  times the natural logarithm of four plus our constant of integration ๐ถ. And now we know the derivative of this expression with respect to ๐‘  will be equal to our integrand. To see this, it might be helpful to rewrite our fundamental theorem of calculus in terms of the variable ๐‘ .

So by using our rules of definite integrals, we found the general antiderivative of our integrand. But we donโ€™t need the general antiderivative. We just need any antiderivative, so we can pick any value of the constant ๐ถ. Weโ€™ll pick ๐ถ equal to zero. This gives us capital ๐น of ๐‘  is equal to one over the natural logarithm of four times ๐‘’ to the power of ๐‘  times the natural logarithm of four. Weโ€™re now ready to start evaluating our definite integral. First, by using our laws of logarithms and our laws of exponents, we were able to rewrite our integrand as ๐‘’ to the power of ๐‘  times the natural logarithm of four. Then, by using the fundamental theorem of calculus, we know this definite integral will be equal to capital ๐น evaluated at four minus capital ๐น evaluated at three.

And itโ€™s worth pointing out you will often see this written in the following form: capital ๐น of ๐‘  inside square brackets with our limits of integration on the side. This is just a short-hand notation for the line above. It helps us keep things neat and tidy. But remember, we already found an expression for our antiderivative capital ๐น of ๐‘ . So weโ€™ll clear some space and write in our expression for capital ๐น of ๐‘ . All thatโ€™s left to do now is evaluate this at the limits of integration. Doing this, we get one over the natural logarithm of four times ๐‘’ to the power of four times the natural logarithm of four minus one over the natural logarithm of four multiplied by ๐‘’ to the power of three times the natural logarithm of four.

And we could use our calculators to evaluate this expression. However, itโ€™s not necessary; we can simplify without. First, weโ€™re going to use the same one of our rules of exponents we did before. However, this time, weโ€™re going to do this in reverse. Weโ€™ll rewrite our first exponent as ๐‘’ to the power of the natural logarithm of four all raised to the fourth power. And weโ€™ll rewrite our second exponent as ๐‘’ to the power of the natural logarithm of four all cubed. But now, by using our laws of logarithms, we know ๐‘’ raised to the power of the natural logarithm of four is just equal to four. So this entire expression simplifies to give us four to the fourth power over the natural logarithm of four minus four cubed over the natural logarithm of four.

But we can simplify this even further. We know that four is equal to two squared. And then by using the power rule for logarithms, we can get that this is equal to two times the natural algorithm of two. So by combining these fractions into one fraction and rewriting our denominator as two times the natural logarithm of two, we get four to the fourth power minus four cubed all divided by two times the natural logarithm of two. And we could keep simplifying in the same way, for example, by taking out the common factor of four cubed in our numerator. However, if we evaluate this expression, we now get 96 divided by the natural logarithm of two. And this is our final answer.

Therefore, by using the fundamental theorem of calculus and what we know about definite integrals of exponential functions, we were able to show the definite integral from three to four of four to the power of ๐‘  with respect to ๐‘  is equal to 96 divided by the natural logarithm of two.

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