Question

1. $\int \frac{40dx}{x^{2}+25}$

Explanation:

Step1: Use integral formula

Recall the integral formula \(\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\arctan(\frac{x}{a})+C\). Here \(a = 5\) and the integral is \(\int\frac{40dx}{x^{2}+25}=40\int\frac{dx}{x^{2}+5^{2}}\).

Step2: Apply the formula

Using the formula \(\int\frac{dx}{x^{2}+5^{2}}=\frac{1}{5}\arctan(\frac{x}{5})+C\), then \(40\int\frac{dx}{x^{2}+5^{2}}=40\times\frac{1}{5}\arctan(\frac{x}{5})+C\).

Step3: Simplify the expression

\(40\times\frac{1}{5}\arctan(\frac{x}{5})+C = 8\arctan(\frac{x}{5})+C\).

Answer:

\(8\arctan(\frac{x}{5})+C\)