Checking Antiderivative Formulas

Verify the f...

Question

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

Accepted Answer

Welcome back everyone. Determine whether the following statement is true or false. The inside derivative of -4 co sequin square of x + 2 divided by 2 is 8 cotangent of x + 2 divided by 2 plus c. A says true and B says false. If this statement is true, then the derivative of a cotangent of x + 2 divided by 2 + c must return the function 4 co sequin square of x + 2 divided by 2. So what we're going to do is simply differentiate. The anti derivative that be the derivative of a cotangent of x plus 2 divided by 2. So to find this derivative we can factor out 8. It is a constant. And differentiate cotangent of x plus 2 divided by 2. Notice that this is a composite function, so we are going to apply the chain rule. So we get 8 multiplied by the derivative of cotangent with respect to x + 2 divided by 2. We call that the derivative of cotangent is negative co sequin squared, so we're multiplying 8 by negative co sequin squared of x + 2. Divided by 2. And now according to the chain rule, we have to multiply it by the derivative of the inner function. So we're multiplying by the derivative of x + 2 divided by 2. And to simplify, we are going to write the previous line as -8 coequin squared of x + 2 divided by 2. Which is now multiplied by the derivative of 1/2 x plus 1. We can basically perform division that's x divided by 2, which is 1/2 x, plus 2 divided by 2, which gives us 1. The derivative of 1 is 0, and the derivative of 1/2 x is 1/2 because the derivative of x is 1. So we basically get -8 multiplied by 1/2 multiplied by. Co sequin squared of x + 2 divided by 2. Right? And this gives us -4. Cosi and squared. Of x plus 2 divided by 2 notice also that we didn't include. The constant c. Why is this the case? Well, the derivative of a constant is 0, right? So when we differentiate the same function plus c, we still get the same steps because we are adding the derivative of c, which is 0. So the constant has no effect. Now notice that we returned the original function, which means that the statement is true. And therefore the correct answer to this problem corresponds to the answer choice A, true. Thank you for watching.

Checking Antiderivative FormulasVerify the formulas in Exercises 57–62 by differentiation.∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

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Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C