$$ I = \int \frac { x d x } { x + a } $$

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The integral $$ \int \frac{x \, dx}{x + a} $$ simplifies to $$ x - a \ln |x + a| + C $$. To solve the integral $$ I = \int \frac{x \, dx}{x + a}, $$ we can use integration by parts or a substitution method. Here, we will use a substitution. Let's rewrite the integrand: $$ \frac{x}{x + a} = 1 - \frac{a}{x + a}. $$ Thus, we can split the integral: $$ I = \int \left( 1 - \frac{a}{x + a} \right) dx. $$ Now, we can integrate each term separately: 1. The integral of $$1$$ is simply $$x$$. 2. For the second term, we have: $$ \int \frac{a}{x + a} \, dx = a \ln |x + a|. $$ Putting it all together, we have: $$ I = x - a \ln |x + a| + C, $$ where $$C$$ is the constant of integration. Thus, the final result is: $$ \int \frac{x \, dx}{x + a} = x - a \ln |x + a| + C. $$

\( I = \int \frac { x d x } { x + a } \)

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\( I = \int \frac { x d x } { x + a } \)

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