evaluate ∫ dx / (25 + x²)².

Explanation:

Step1: Use trig - substitution

Let $x = 5\tan\theta$, then $dx=5\sec^{2}\theta d\theta$. And $25 + x^{2}=25+25\tan^{2}\theta = 25\sec^{2}\theta$.

Step2: Substitute into the integral

The integral $\int\frac{dx}{(25 + x^{2})^{2}}$ becomes $\int\frac{5\sec^{2}\theta d\theta}{(25\sec^{2}\theta)^{2}}=\int\frac{5\sec^{2}\theta d\theta}{625\sec^{4}\theta}=\frac{1}{125}\int\cos^{2}\theta d\theta$.

Step3: Use the double - angle formula

Since $\cos^{2}\theta=\frac{1 + \cos(2\theta)}{2}$, the integral $\frac{1}{125}\int\cos^{2}\theta d\theta=\frac{1}{125}\int\frac{1+\cos(2\theta)}{2}d\theta=\frac{1}{250}\int(1+\cos(2\theta))d\theta$.

Step4: Integrate term - by - term

$\frac{1}{250}\int(1+\cos(2\theta))d\theta=\frac{1}{250}(\theta+\frac{1}{2}\sin(2\theta))+C$.

Step5: Back - substitute

Since $x = 5\tan\theta$, then $\theta=\arctan(\frac{x}{5})$ and $\sin(2\theta)=\frac{2\tan\theta}{1 + \tan^{2}\theta}=\frac{2\frac{x}{5}}{1+\frac{x^{2}}{25}}=\frac{10x}{25 + x^{2}}$. So the integral is $\frac{1}{250}(\arctan(\frac{x}{5})+\frac{x}{25 + x^{2}})+C$.

Answer:

$\frac{1}{250}(\arctan(\frac{x}{5})+\frac{x}{25 + x^{2}})+C$