Question

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int \frac{1}{1+u^{2}} d u=\tan ^{-1} u+C$$

Short answer

By taking the derivative of the result of the integral \(\tan^{-1}(u) + C\) and comparing it to the integrand \(\frac{1}{1+u^2}\), we confirm that the given integral formula holds true.

Step-by-step solution

Understand the problem

We are given the integral \(\int \frac{1}{1+u^{2}} d u=\tan ^{-1} u+C\) and are asked to verify this equality using differentiation. If we take derivative of the right side of the equation, we should end up with the integrand on the left side, as per the Fundamental Theorem of Calculus.

Differentiate the antiderivative

Differentiate \(\tan^{-1}(u) + C\) using the chain rule. The derivative of \(\tan^{-1}(u)\) is \(\frac{1} {1+u^2} \), and the derivative of a constant \(C\) is zero. Thus, the derivative of \(\tan^{-1}(u) + C\) is \(\frac{1} {1+u^2} \).

Verify the equality

Comparing this result with the integrand of the given integral, we notice that they are identical. This confirms that \(\tan^{-1}(u) + C\) is indeed an antiderivative of \(\frac{1}{1+u^2}\).