Video: Integrating Non-Standard Exponential Functions

Find ∫ 2(5^(4π‘₯)) dπ‘₯.

04:28

Video Transcript

Find the integral of two multiplied by five raised to the power of four π‘₯ with respect to π‘₯.

The first thing we notice about this question is that our integrand is not in a standard form which we know how to integrate. However, we can notice that five raised to the power of four π‘₯ is an exponential function. And these exponential functions we do know how to integrate. For example, for constant π‘Ž and a constant 𝑛 not equal to zero. Then the integral of π‘Ž multiplied by 𝑒 to the power of 𝑛 multiplied by π‘₯ with respect to π‘₯ is equal to π‘Ž divided by 𝑛 multiplied by 𝑒 to the power of 𝑛π‘₯ plus the constant of integration 𝑐.

So if we can manipulate our integrand of two multiplied by five to the power of four π‘₯ to be in the form of π‘Ž multiplied by 𝑒 to the power of 𝑛π‘₯, then we can use this rule to integrate. To do this, we recall one of our rules of exponents, which tells us that π‘Ž raised to the power of 𝑏 multiplied by 𝑐 is the same as raising π‘Ž to the power of 𝑏 and then raising this all to the power of 𝑐. So if we use this on five raised to the power of four multiplied by π‘₯, we get five raised to the power of four. And then we raise all of this to the power of π‘₯.

This is now almost in a form which we can integrate. We just want instead of five to the fourth power, we want 𝑒 raised to the 𝑛th power. So to do this, we want to solve the equation five to the fourth power is equal to 𝑒 to the 𝑛th power. We can do this by taking logs of both sides. We get the natural log of five to the fourth power is equal to the natural log of 𝑒 to the 𝑛th power.

We can simplify this by using our log laws. We have that the log of 𝑏 to the power of 𝑐 is equal to 𝑐 multiplied by the log of 𝑏. So we can simplify the natural log of five to the fourth power to be equal to four multiplied by the natural log of five. And we can also simplify the natural log of 𝑒 to the 𝑛th power to just be equal to 𝑛. So what we’ve shown is that 𝑒 raised to the power four multiplied by the natural logarithm of five is equal to five raised to the fourth power. So we can substitute this into our manipulation. So we’ve shown that five raised to the power of four π‘₯ is equal to 𝑒 raised to the power of four multiplied by the natural logarithm of five all raised to the power of π‘₯.

There’s one more piece of manipulation that we want to do, since we want this to be of the form 𝑒 to the power of π‘Žπ‘₯, where π‘Ž is a constant. We’re going to use our laws of exponents again. Since π‘Ž raised to the power of 𝑏 all raised to the power of 𝑐 is equal to π‘Ž raised to the power of 𝑏 multiplied by 𝑐. We have that 𝑒 raised to the power of four multiplied by the natural logarithm of five all raised to the power of π‘₯ is equal to 𝑒 raised to the power of four multiplied by the natural logarithm of five multiplied by π‘₯. So we’ve successfully manipulated five to the power of four π‘₯ to be in the form of 𝑒 to the power of π‘Žπ‘₯.

So we’re now ready to evaluate our integral. And we’ll start by substituting in our expression for five raised to the power of four π‘₯. This gives us the integral of two multiplied by 𝑒 to the power of four multiplied by the natural logarithm of five multiplied by π‘₯ with respect to π‘₯. Since four multiplied by the natural logarithm of five and two are both constants, we can use our integral law to evaluate this integral. This gives us two divided by four multiplied by the natural logarithm of five. And then we multiply all of this by 𝑒 to the power of four multiplied by the natural logarithm of five multiplied by π‘₯. And then we add our constant of integration 𝑐.

Now we could stop here, but we can simplify this further. We can start by canceling the shared factor of two in the numerator and the denominator. We recall that 𝑒 raised to the power of four multiplied by the natural logarithm of five multiplied by π‘₯ is exactly equal to five raised to the power of four π‘₯. This gives us five raised to the power of four π‘₯ divided by two multiplied by the natural algorithm of five plus the constant of integration 𝑐.

Finally, we can simplify two multiplied by the natural logarithm of five by using one of our log laws. Using this law would give us that two multiplied by the natural logarithm of five is equal to the natural logarithm of five squared, which we can evaluate to be equal to the natural logarithm of 25. This gives us that the integral of two multiplied by five raised to the power of four π‘₯ with respect to π‘₯ is equal to. Five raised to the power of four π‘₯ divided by the natural logarithm of 25 plus the constant of integration 𝑐.

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