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Chapter 7: Problem 39

Evaluate the following integrals. $$\int \frac{x^{2}}{\left(25+x^{2}\right)^{2}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral: $$\int \frac{x^2}{(25+x^2)^2} dx$$ Answer: $$-\frac{1}{2(25+x^2)} + C$$

Step by step solution

01

Identify a substitution

Let's use the substitution: $$u = 25 + x^2$$
02

Differentiate and find dx

Now differentiate u with respect to x to find dx: $$\frac{d u}{d x} = 2x$$ Solving for dx, we get: $$d x=\frac{d u}{2 x}$$
03

Rewrite the integral in terms of u

We can now rewrite the integral as follows: $$\int \frac{x^2}{\left(25+x^2\right)^2} dx = \int\frac{1}{u^2} \cdot x^2 \cdot \frac{du}{2x} = \frac{1}{2} \int \frac{1}{u^2} du$$
04

Integrate with respect to u

Now, we will integrate with respect to u: $$\frac{1}{2}\int \frac{1}{u^2} d u = -\frac{1}{2u} + C$$
05

Replace u with the original variable x

Finally, we'll rewrite the result in terms of the original variable x, using our substitution: $$-\frac{1}{2u} + C = -\frac{1}{2(25+x^2)} + C$$ So, the final answer is: $$\int \frac{x^{2}}{\left(25+x^{2}\right)^{2}} d x=-\frac{1}{2(25+x^2)} + C$$

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