Question

Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{a^{2}+u^{2}}=\frac{1}{a} \arctan \frac{u}{a}+C $$

Short answer

The derivative of \(\frac{1}{a} \arctan(\frac{u}{a}) + C\) is indeed \(\frac{1}{a^{2}+u^{2}}\), which verifies that the integral formula is correct.

Step-by-step solution

Prepare the right side of the equation for differentiation

The right side of the equation is \( \frac{1}{a} \arctan \frac{u}{a}+C \). We'll differentiate this with respect to \(u\).

Apply the Constant Multiple Rule

Differentiation of the first part, that is, \( \frac{1}{a} \arctan \frac{u}{a} \). We take out \(\frac{1}{a}\) from the derivate because of the constant multiplication rule in differentiation, which means the derivative of a constant times a function is the constant times the derivative of the function. So, it becomes \(\frac{1}{a} (\frac{d}{du} \arctan \frac{u}{a})\). The derivative of the constant \(C\) which is the second part, is 0, because the derivative of a constant is always zero.

Use the Derivative Rule for Inverse Tangent

Now, we have to differentiate the inverse tangent function. The derivative of inverse tangent of \(x\) is \(\frac{1}{1+x^{2}}\), so the derivative of \(\arctan \frac{u}{a}\) will be \(\frac{1}{1+(\frac{u}{a})^{2}}\). Hence, the derivative becomes \( \frac{1}{a} \cdot \frac{1}{1+(\frac{u}{a})^{2}}\).

Simplify the expression

We can simplify this expression by multiplying both numerator and denominator by \(a^{2}\), resulting in \(\frac{1}{a^{2}+u^{2}}\), which is the integrand on the left. Thus, the derivative of the given expression equals the integrand on the left, demonstrating that the differentiation was performed successfully