Problem 61

Question

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

Step-by-Step Solution

Verified

Answer

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

1Step 1: 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.

2Step 2: 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.

3Step 3: 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} $

Problem 61

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