Question

evaluate the integral \\( \int \frac{dy}{\left( \arctan \left( \frac{y}{7} \
ight) \
ight) \left( 49 + y^2 \
ight)} \\).

Explanation:

Step1: Use substitution

Let \(u = \arctan(\frac{y}{7})\). Then \(du=\frac{1}{1 + (\frac{y}{7})^2}\cdot\frac{1}{7}dy=\frac{7}{49 + y^{2}}dy\), and \(\frac{dy}{(49 + y^{2})}=\frac{du}{7}\).

Step2: Rewrite the integral

The original integral \(\int\frac{dy}{(\arctan(\frac{y}{7}))(49 + y^{2})}\) becomes \(\frac{1}{7}\int\frac{du}{u}\).

Step3: Integrate

We know that \(\int\frac{du}{u}=\ln|u|+C\). So \(\frac{1}{7}\int\frac{du}{u}=\frac{1}{7}\ln|u|+C\).

Step4: Substitute back

Substituting \(u = \arctan(\frac{y}{7})\) back, we get \(\frac{1}{7}\ln|\arctan(\frac{y}{7})|+C\).

Answer:

\(\frac{1}{7}\ln|\arctan(\frac{y}{7})|+C\)