Question

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int \csc ^{2} u d u=-\cot u+C$$

Short answer

The verification of the antiderivative formula, \( \int \csc ^{2} u du = -\cot u+C \), is achieved by differentiating \(-\cot u+C\) to receive the original function \(\csc ^{2} u\).

Step-by-step solution

Write down the provided integral and its equivalent

The provided integral is \( \int \csc ^{2} u du \) and it is given that this is equivalent to \(-\cot u+C\).

Differentiating the obtained function

To verify the antiderivative formula, differentiate the obtained function which is \(-\cot u+C\). The derivative of \(-\cot u\) is \(\csc ^{2} u\) and the derivative of \(C\) is \(0\) (since \(C\) is a constant). Therefore, the derivative of \(-\cot u+C\) is \(\csc ^{2} u\).

Comparing the functions

As the derivative of \(-\cot u+C\) is \(\csc ^{2} u\) which is the original function. Thus, this verifies the antiderivative formula.