Question

State the integration formula you would use to perform the integration. Do not integrate. $$ \int \frac{x}{x^{2}+4} d x $$

Short answer

The integration formula with respect to \( u \) would be \( \int \frac{1}{u} * \frac{du}{2} \).

Step-by-step solution

Identify Integral Type

The given function \( \int \frac{x}{x^{2}+4} dx \) is the integral of a rational function where the power of the numerator is less than the power of the denominator, therefore a good strategy may be to use substitution.

Choose a Substituion Variable

Choose a variable to substitute that, when differentiated, appears in the rest of the function to simplify the integral. The obvious choice is \( u = x^2 + 4 \) since its derivative \( du = 2x dx \) is present in the rest function.

State the Integration Formula

We need to express the integration in terms of \( u \). It's better to find \( dx \) in terms of \( du \). Dividing both sides of \( du = 2x dx \) by \( 2x \), we get \( dx = \dfrac{du}{2x} \). Now, substitute \( u \) and \( dx \) in the original integral, so the integration formula becomes: \( \int \frac{1}{u} * \frac{du}{2} \)