Finding Antiderivatives
In Exercises 1–16, find an antid...

Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

2 - 5 / x²

Accepted Answer

Welcome back everyone. Find an anti derivative for the function 3 minus 8 divided by x2. A says 3 x + 8 divided by x + c. B 3 divided by x + 8 divided by x2 + c. C 3 x + 8 + C. and D x divided by 3 + 8 divided by x plus c. For this problem we want to find the derivative F of X, whose derivative is the function 3 minus 8 divided by x 2. We can also express it as 3 minus 8 multiplied by x to the power of -2, which suggests that we can apply the power rule for differentiation. Let's remember 2 rules. The derivative of C multiplied by x is equal to C, right? This is how we can get our constant, and similarly, we can apply the power rule in a general form. To show that the derivative of c multiplied by x to the power of a is equal to c multiplied by a multiplied by x to the power of a minus 1. So from the first rule, if we equate c equals 3. We can show that the derivative of 3 x is equal to 3. Additionally, we can add a constant capital C to show that the derivative of 3 x plus c is also equal to 3. So. We are going to get the whole family of anti derivatives later. For now, it's just sufficient to show that the antiderivative of 3 x is 3, and we can add C later, right, when we get all of the antiderivatives. And now we can identify the inside derivative of -8. X to the power of -2. Using our next rule, we can first of all, by analogy, compare the exponents. And show that A minus 1 is equal to -2, which gives us A equals -2 + 1, that's -1. And similarly, C multiplied by a is our constant, which is -8. So if c multiplied by a is equals -8. Then C must be equal to -8 divided by A, which is -8 divided by -1, and that's 8. Therefore, the derivative of 8th X of -1 is equal to -8, X of -2, right? We can also write it as 8 divided by x. Derivative is equal to -8, x to the power of -2. And now we can combine results 1 and 2. To show that the derivative of. We X Plus 8 divided by x plus we can add some constant c because the derivative of a constant is 0. The derivative of this function is 3 minus. 8 divided by x 2. Therefore, the correct answer for this problem corresponds to the answer choice a. Thank you for watching.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.2 - 5 / x²

Watch next

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
2 - 5 / x²