∫ (x² + 4) / x² dx

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Understand the Problem

The question is asking to evaluate the integral of the function (x² + 4) / x² with respect to x. To solve this, we'll rewrite the expression and perform integration.

Answer

The integral evaluates to $x - \frac{4}{x} + C$.
Answer for screen readers

The final answer for the integral is: $$ x - \frac{4}{x} + C $$

Steps to Solve

  1. Rewrite the Integral

Start by simplifying the integrand: $$ \frac{x^2 + 4}{x^2} = \frac{x^2}{x^2} + \frac{4}{x^2} = 1 + \frac{4}{x^2} $$ Thus, the integral becomes: $$ \int \left(1 + \frac{4}{x^2}\right) dx $$

  1. Separate the Integral

Now, separate the integral for easier computation: $$ \int \left(1 + \frac{4}{x^2}\right) dx = \int 1 , dx + \int \frac{4}{x^2} , dx $$

  1. Integrate Each Term

Now, compute the integrals separately:

  • The integral of $1$ is: $$ \int 1 , dx = x $$
  • The integral of $\frac{4}{x^2}$ can be rewritten as: $$ \int \frac{4}{x^2} , dx = 4 \int x^{-2} , dx = 4 \left(-\frac{1}{x}\right) = -\frac{4}{x} $$
  1. Combine the Results

Now, combine the results of the integrals: $$ x - \frac{4}{x} + C $$ where $C$ is the constant of integration.

The final answer for the integral is: $$ x - \frac{4}{x} + C $$

More Information

The integral evaluates a function over the variable $x$. This involves breaking down the function and integrating each part individually, which is a common technique in calculus.

Tips

  • Not simplifying the integrand: Always simplify the function before integrating to make the process easier.
  • Forgetting the constant of integration: It's essential to include the constant $C$ when performing indefinite integrals.
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