Question Video: Integration of a Power Function and a Reciprocal Function Mathematics • Higher Education

Determine ∫ (8π‘₯⁹ + 4/π‘₯) dπ‘₯.

02:25

Video Transcript

Determine the integral of eight multiplied by π‘₯ to the ninth power plus four divided by π‘₯ with respect to π‘₯.

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

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

We can use this to integrate eight multiplied by π‘₯ to the ninth power with respect to π‘₯. We’ll set our exponent, 𝑛, equal to nine and our coefficient, π‘Ž, equal to eight. Doing this gives us eight divided by nine plus one multiplied by π‘₯ 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 π‘Ž divided by π‘₯ with respect to π‘₯ is equal to π‘Ž multiplied by the natural logarithm of the absolute value of π‘₯ plus a constant of integration, 𝑐. We can use this to integrate four divided by π‘₯ with respect to π‘₯. We’ll set our coefficient, π‘Ž, to be equal to four. This gives us four multiplied by the natural logarithm of the absolute value of π‘₯ 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 π‘₯ 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 π‘₯ to the 10th power plus 𝑐 plus four multiplied by the natural logarithm of the absolute value of π‘₯. 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 π‘₯ to the ninth power plus four over π‘₯ with respect to π‘₯ is equal to four multiplied by π‘₯ to the 10th power divided by five plus four multiplied by the natural logarithm of the absolute value of π‘₯ plus a constant of integration, 𝑐.

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