d) $$ \int \frac{x+4}{ ext { solution: }} \frac{x^{2}-6 x+25}{} d x $$

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d) $$ \int \frac{x+4}{	ext { solution: }} \frac{x^{2}-6 x+25}{} d x $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int \frac{x+4}{x^{2}-6x+25} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{x+4}{x^{2}-6x+25} dx$$
- step1: Rewrite the fraction:
  $$\int \frac{x-3}{x^{2}-6x+25}+\frac{7}{x^{2}-6x+25} dx$$
- step2: Use properties of integrals:
  $$\int \frac{x-3}{x^{2}-6x+25} dx+\int \frac{7}{x^{2}-6x+25} dx$$
- step3: Evaluate the integral:
  $$\frac{1}{2}\ln{\left(\left|x^{2}-6x+25\right|\right)}+\int \frac{7}{x^{2}-6x+25} dx$$
- step4: Evaluate the integral:
  $$\frac{1}{2}\ln{\left(\left|x^{2}-6x+25\right|\right)}+\frac{7}{4}\arctan\left(\frac{1}{4}x-\frac{3}{4}\right)$$
- step5: Add the constant of integral C:
  $$\frac{1}{2}\ln{\left(\left|x^{2}-6x+25\right|\right)}+\frac{7}{4}\arctan\left(\frac{1}{4}x-\frac{3}{4}\right) + C, C \in \mathbb{R}$$
The integral of $$ \frac{x+4}{x^{2}-6x+25} $$ with respect to $$ x $$ is given by:

$$ \frac{1}{2}\ln{|x^{2}-6x+25|} + \frac{7}{4}\arctan\left(\frac{1}{4}x-\frac{3}{4}\right) + C $$

where $$ C $$ is an arbitrary constant.
The integral of $$ \frac{x+4}{x^{2}-6x+25} $$ with respect to $$ x $$ is: $$ \frac{1}{2}\ln{|x^{2}-6x+25|} + \frac{7}{4}\arctan\left(\frac{1}{4}x-\frac{3}{4}\right) + C $$ where $$ C $$ is a constant.

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