Question Video: Finding the Indefinite Integral of a Reciprocal Function Mathematics

Determine β« β(2/7π₯) dπ₯.

02:02

Video Transcript

Determine the indefinite integral of negative two divided by seven π₯ with respect to π₯.

In this question, weβre asked to evaluate the indefinite integral of a reciprocal function. And we can do this directly by recalling one of our integral results. For any real constant π, the indefinite integral of π over π₯ with respect to π₯ is equal to π times the natural logarithm of the absolute value of π₯ plus the constant of integration πΆ. We can just note that our value of π will be negative two over seven; however, sometimes it can be difficult to see this. So, instead, weβll just take the constant factor of negative two over seven outside of our integral. This leaves us with negative two over seven times the indefinite integral of the reciprocal function, one over π₯ with respect to π₯.

And we know the integral of the reciprocal function is the natural logarithm of the absolute value of π₯. We can just recall this, or we can set our value of π equal to one into our integral result. Using either method, weβve shown the indefinite integral of negative two over seven π₯ with respect to π₯ is negative two-sevenths times the natural logarithm of the absolute value of π₯ plus πΆ. And itβs worth noting we can always check our answer by using differentiation. Remember, when weβre finding the indefinite integral of a function, weβre finding its most general antiderivative. This means when we differentiate our function with respect to π₯, we should end up with our integrand.

So, letβs evaluate the derivative of negative two-sevenths times the natural logarithm of the absolute value of π₯ plus πΆ with respect to π₯. We can start by recalling the derivative of the natural logarithm of the absolute value of π₯ with respect to π₯ is one over π₯. So, when we differentiate the first term with respect to π₯, we get negative two over seven multiplied by one over π₯. The second term is a constant, so its rate of change with respect to π₯ is zero. This just leaves us with negative two-sevenths times one over π₯, which we can simplify is negative two over seven π₯. This is our integrand, so this confirms that our answer is correct. Therefore, the indefinite integral of negative two over seven π₯ with respect to π₯ is negative two over seven times the natural logarithm of the absolute value of π₯ plus πΆ.