Video Transcript
Determine the integral of the natural logarithm of five divided by four 𝑥 with respect to 𝑥.
The question wants us to determine an indefinite integral. The first thing we should do is look at our integrand. We can see the numerator of our integrand is the natural logarithm of five. This is a positive constant. And the denominator of our integrand is four 𝑥. And four is also a constant. So, we want to take the constant coefficient of the natural logarithm of five divided by four outside of our integral.
And we can do this because we know for any constant 𝑘, the integral of 𝑘 times 𝑓 of 𝑥 with respect to 𝑥 is equal to 𝑘 times the integral of 𝑓 of 𝑥 with respect to 𝑥. So, applying this to our integral with 𝑘 equal to the natural logarithm of five over four, which we’ll rewrite as one-quarter times the natural logarithm of five. We get that our integral is equal to a quarter times the natural logarithm of five times the integral of one over 𝑥 with respect to 𝑥.
In other words, we just need to find the integral of the reciprocal function with respect to 𝑥. And we know how to integrate the reciprocal function with respect to 𝑥. It’s equal to the natural logarithm of the absolute value of 𝑥 plus a constant of integration 𝑐. So, applying this to our integral, we get one-quarter times the natural logarithm of five multiplied by the natural logarithm of the absolute value of 𝑥 plus a constant of integration we’ll call 𝑐 one.
All that’s left now is to simplify our answer. We’ll start by distributing a quarter times the natural logarithm of five over our parentheses. Doing this, we get one-quarter times the natural logarithm of five multiplied by the natural logarithm of the absolute value of 𝑥 plus 𝑐 one over four times the natural logarithm of five.
And we’ll do one last thing to simplify this expression. Since 𝑐 one is a constant, 𝑐 one divided by four multiplied by the natural logarithm of five is also a constant. So, we’ll just write this is as a new constant we’ll call 𝑐. And this gives us our final answer. The integral of the natural logarithm of five divided by four 𝑥 with respect to 𝑥 is equal to one-quarter times the natural logarithm of five multiplied by the natural logarithm of the absolute value of 𝑥 plus 𝑐.
