Question Video: Finding the Indefinite Integral of an Exponential Function with an Integer Base

a nice integral and an important ignored constant
a nice integral and an important ignored constant

Video Transcript

Determine the indefinite integral
of two to the power of nine 𝑥 with respect to 𝑥.

In this question, we’re asked to
evaluate the integral of an exponential function. And we can note this looks very
similar to one of our integral results. For any positive constant 𝑎 not
equal to one, the integral of 𝑎 to the power of 𝑥 with respect to 𝑥 is equal to
𝑎 to the power of 𝑥 divided by the natural logarithm of 𝑎 plus the constant of
integration 𝐶. However, in this case, our exponent
is not just equal to 𝑥; we have nine 𝑥. So, we can’t just directly apply
this integral result. And there are two ways we can try
and fix this. We could try using the
𝑢-substitution. 𝑢 is nine 𝑥. This would then allow us to rewrite
our integral in this form. However, there’s actually a simpler
method.

We can just apply our laws of
exponents. Two to the ninth power all raised
to the power of 𝑥 will be equal to two to the power of nine times 𝑥. So, we can rewrite our integral as
the integral of two to the ninth power all raised to the power of 𝑥 with respect to
𝑥. Then, our value of 𝑎 is two to the
ninth power. We could now evaluate two to the
ninth power as 512. However, we’re just going to
substitute 𝑎 is equal to two to the power of nine into our integral result. This then gives us two to the ninth
power all raised to the power of 𝑥 all divided by the natural logarithm of two to
the ninth power plus the constant of integration 𝐶. And this can help us see why it’s
useful to leave this as two to the ninth power rather than evaluate.

We can simplify the numerator by
using our laws of exponents, and we can simplify our denominator by using the power
rule for logarithms. First, in the numerator, two raised
to the power of nine all raised to the power of 𝑥 will be two to the power of nine
times 𝑥. Next, in the denominator, the power
rule for logarithms tells us the natural logarithm of two to the ninth power will be
nine times the natural logarithm of two, which then gives us our final answer. The integral of two to the power of
nine 𝑥 with respect to 𝑥 is two to the power of nine 𝑥 divided by nine times the
natural logarithm of two plus the constant of integration 𝐶.

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