Finding antiderivatives. Find all the antider...

Question

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

ƒ(x) = 2 sinx + 1

Accepted Answer

Welcome back, everyone. Find all the inside derivatives of the function F of X equals -5 x + 2. Now let's suppose that our anti derivatives correspond to F of X. If we differentiate F of X, we must get lowercase f x, right? In other words, we're looking for the antiderivatives F of X that satisfy this relationship, and we know that capital F. of x must be equal to -5 x + 2. So now we want to think of differentiation and simply go back. We go in reverse to identify the family of functions F of X, and as we can see on the right hand side, we have two separate functions. One of them involves sine. So what we have to do is simply remember that the derivative of cosine of X. Is equal to negative sign. Of X and the second function that we're considering is a constant. It is 2, right? So now if we focus on the elementary function X, let's recall that the derivative of X is equal to 1 based on the power rule, and then we're going to use these two relationships to arrive at the final answer. Considering -5 sin of x and the fact that the derivative of cosine of X is a negative sign of X, we can simply multiply both sides by 5. So we get 5 multiplied by the derivative of cosine of X equals -5 sin of x, right? So we can clearly see that. The derivative of 5 cosine of X. In other words, we're incorporating our constant into our function because we can always factor out a constant from the derivative. The derivative of 5 cosine of X is equal to -5 sin of x, so we have the first part of our answer. Our family of nine derivatives is going to have 5 cosine of X, and now let's focus on the fact that the derivative of X is 1. We can multiply both sides by 2. So we can show that the derivative of 2 X is equal to 2, and this is our second function within the family of inside derivatives. All that we have to do is simply combine them. We can now show that F. Capital F I'm sorry of X is equal to the derivative of 5 cosine of x. Plus the derivative of 2 X. And finally, because we're looking for all the inside derivatives, let's recall that the derivative of any constant C is zero. So we're also going to add a constant C. And now let's notice that both hands sides contain the derivative, right, meaning the capital F of X is simply equal to 5 cosine of X. Plus 2 x + C and that would be our final answer for this problem. Thank you for watching.

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

ƒ(x) = 2 sinx + 1

Key Concepts

Antiderivative

Basic Integration Rules

Verification by Differentiation

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