Problem 60 - Essential Calculus

Question

State the integration formula you would use to perform the integration. Do not integrate. $$ \int \frac{x}{\left(x^{2}+4\right)^{3}} d x $$

Step-by-Step Solution

Answer

The integration formula to be used is \( \frac{1}{2} \int \frac{1}{u^{3}} du \), where \( u = x^{2}+4 \).

Step 1: Identification of given elements

The integrand function is \( \frac{x}{\left(x^{2}+4\right)^{3}} \). Identify the function inside the brackets and its derivative.

Step 2: Formulate substitution

Looking at the integral, notice that the numerator is the derivative of the denominator except for a constant factor. Hence, a suitable substitution would be to set \( u = x^{2}+4 \). Then, the derivative \( du = 2x dx \). To make this substitution work, divide both sides by 2 to get \( \frac{1}{2} du = x dx \) which correlates with the integral we need to perform.

Step 3: Final substitute integral

Substitute \( x dx \) and \( \left(x^{2}+4\right)^{3} \) with \( \frac{1}{2} du \) and \( u^{3} \) respectively in the integrand. The integral then becomes \( \frac{1}{2} \int \frac{1}{u^{3}} du \).