Question Video: Integration of a Power Function and a Reciprocal Function Mathematics

Video Transcript

Determine the integral of eight multiplied by 𝑥 to the ninth power plus four divided by 𝑥 with respect to 𝑥.

We start by recalling that the integral of the sum of two functions 𝑓 and 𝑔 with respect to 𝑥 is equal to the integral of 𝑓 with respect to 𝑥 plus the integral of 𝑔 with respect to 𝑥. So, we can use this to split our integral into two separate integrals. We get the integral of eight multiplied by 𝑥 to the ninth power with respect to 𝑥 plus the integral of four divided by 𝑥 with respect to 𝑥.

Next, we recall that if 𝑛 is not equal to negative one, then the integral of some constant 𝑎 multiplied by 𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑎 divided by 𝑛 plus one multiplied by 𝑥 to the power of 𝑛 plus one plus our constant of integration, 𝑐.

We can use this to integrate eight multiplied by 𝑥 to the ninth power with respect to 𝑥. We’ll set our exponent, 𝑛, equal to nine and our coefficient, 𝑎, equal to eight. Doing this gives us eight divided by nine plus one multiplied by 𝑥 to the power of nine plus one plus a constant of integration we will call 𝑐 one.

Now, we recall that the integral of some constant 𝑎 divided by 𝑥 with respect to 𝑥 is equal to 𝑎 multiplied by the natural logarithm of the absolute value of 𝑥 plus a constant of integration, 𝑐. We can use this to integrate four divided by 𝑥 with respect to 𝑥. We’ll set our coefficient, 𝑎, to be equal to four. This gives us four multiplied by the natural logarithm of the absolute value of 𝑥 plus a constant of integration we will call 𝑐 two.

We’re now ready to start simplifying. We have nine plus one is equal to 10. So, our first term is eight divided by 10 multiplied by 𝑥 to the 10th power. We have the both 𝑐 one and 𝑐 two, our constants of integration, so we can combine both of these into a new constant, which we will call 𝑐. This gives us eight over 10 multiplied by 𝑥 to the 10th power plus 𝑐 plus four multiplied by the natural logarithm of the absolute value of 𝑥. Finally, we can simplify eight divided by 10 to just be four divided by five.

Therefore, we have shown that the integral of eight multiplied by 𝑥 to the ninth power plus four over 𝑥 with respect to 𝑥 is equal to four multiplied by 𝑥 to the 10th power divided by five plus four multiplied by the natural logarithm of the absolute value of 𝑥 plus a constant of integration, 𝑐.