Give the antiderivatives of xᵖ . For what values of p does you...

Question

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

Accepted Answer

Welcome back, everyone. Let p be a constant. Find the family of an inside derivatives F of X of the function f of x equals 5x to the power of P and state the values of P for which the answer is valid. So for this problem, if F of X corresponds to the family of anti derivatives, this means that the derivative of F of X must be equal to lower f of X, right? And we want to identify F of X. Now what we can see is that lowercase f of X is 5 multiplied by x raises the power of p. So specifically we're considering a function in the form of X or race to some constant, and therefore we want to remember the power rule. Let's recall that the derivative of a constant c multiplied by x to the power of n. is equal to the constant C multiplied by x the power of N minus 1, right? Multiplied by N. So we can write it as C multiplied by N multiplied by X to the power of N minus 1. So, what we want to do is simply equate this to 5 multiplied by X, erase the power of B, and basically identify our constant C and N, right? Then we will be able to define our capital F of X. What we're going to do is simply understand that our exponent N minus 1 is equal to p because this is the only exponent for X. And from here, we can solve for N. N is going to be equal to P + 1. We can add we can add 1 to both sides. So we have successfully identified N. And now we can identify the constant sea. C multiplied by n, which is P + 1. Must be equal to 5 because this is the constant on the right hand side. So solving for C we get C equals 5 divided by. P + 1 Well done, so we have an expressed in terms of P and we have C in terms of P, meaning we can now successfully identify the family of anci derivatives. Now F of X is going to be equal to 5 divided by P plus 1. That's our constant see multiplied by. X To the power of N, right? So essentially we're multiplying by Xa to the power of P + 1. Now what we have to do is understand that this is one of the possible functions. Let's assume that we add some constant C. Remember that the derivative of a constant is 0, so when we differentiate this family of functions F of X, we're going to get. The function lowercase F of X, right? Because the derivative of any constant is zero. Therefore, we have successfully identified the family of inside derivatives, and now we have to say that. The values of P for which the answer is valid corresponds to p + 1 not being equal to 0 because division by zero is undefined, meaning P is not equal to -1. We can subtract one from both sides. So, basically speaking, our first answer is that the family of anti derivatives is 5 divided by P + 1 multiplied by X, raise the power of P + 1 plus some constant C, where P is not equal to -1. That would be our final answer, and thank you for watching.

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

Key Concepts

Antiderivative

Power Rule for Integration

Domain of the Antiderivative

Watch next