Finding antiderivatives. Find all the antider...

Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(y) = - 2/y³

Accepted Answer

Welcome back, everyone. What is the general anci derivative of P of C equals 8 divided by Z2d? Now, let's identify the general an inside derivative and to begin with, let's label it as capital P of Z. If this is the antiderivative of lowercase b of c, then the derivative of capital P ofc must be equal to. The lowercase POC. This is the relationship that we are going to use. So the derivative of capital P of Z must be equal to 8 divided by Z2d. Using the reciprocal rule, we can also write this form as 8 multiplied by Z raises the power of -2. And what we can see is that our Z has an exponent which is a constant, meaning we can use the power rule. Let's recall the power rule. If we have a constant C multiplied by the variable C erase the power of N, this derivative is going to be equal to the constant C multiplied by N multiplied by Z raise the power of N minus 1. So let's assume that P of C and specifically capital P of Z has a form of C multiplied by. Z raised the power of M. Our goal is to identify C and N, right? So from here, if we consider the right hand side, we can show that our N minus 1, the final exponent, is equal to -2. When we say the final exponent, this means the exponent that we get after differentiation. So N minus 1 must be equal to -2. Meaning we can solve for n and is going to be equal to -2 + 1, which is -1. And finally we can focus on. C multiplied by n which corresponds to the constant 8 in front of Z, right? So C multiplied by M, which is C multiplied by 1 or negative C, must be equal to 8. Therefore, solving this equation, we can show that C is equal to -8. Well then, so we have identified the constant C and the exponent n. We have shown that C is -8. And Is equal to -1. Therefore p of C. Must be equal to -8 Zera to the power of -1. We're simply negative 8 divided by z if we apply the reciprocal rule. However, we're looking for the general expression of the antiderivative. So when we differentiate P of C, we indeed get 8 divided by Z squared, but also let's recall that if we differentiate some constant C, the derivative of a constant is 0, so. Adding a constant C to our capital P of Z. Essentially it gives us a family of anti derivatives, right? So no matter what constant we add. The differentiation of negative 8 divided by Z + C will always lead to 8 divided by Z2, so adding this constant is essential. Therefore, our final answer for this problem is -8 divided by Z + C. Thank you for watching.

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(y) = - 2/y³

Key Concepts

Antiderivative

Power Rule for Integration

Verification by Differentiation

Watch next