Problem 23 - Essential Calculus

Question

Find the integral. $$ \int \frac{-5 x}{\left(x^{2}+5\right)^{3 / 2}} d x $$

Step-by-Step Solution

Answer

The solution to the integral is \(5/\sqrt{x^2 + 5} + C\).

Step 1: Define the Substitution Variable

Define a new variable, \(u\), to simplify the integral. A good choice here is \(u = x^2 + 5\), since \(u\) appears in the denominator and its derivative, \(du/dx=2x\), is part of the numerator.

Step 2: Find the Derivative of u

Now find the derivative, \(du/dx\), which will replace \(x\) in the integral. Here, \(du/dx = 2x\). Multiplying both sides by \(dx\) gives \(du = 2x dx\).

Step 3: Rearrange the Equation for dx

Rearrange the equation from step 2 to solve for \(dx\). This gives \(dx=du/(2x)\).

Step 4: Substitute u and dx into the Integral

Now that we have expressions for \(u\) and \(dx\), substitute them into the integral. This will change the integral to \(\int (−5 / 2) (1/u^{3/2}) du\).

Step 5: Solve the Integral

Now solve this new, simpler integral. That is, the integral is \(-(5/2) \int (1/u^{3/2}) du = -(5/2)(-2/(\sqrt u)) + C = 5/\sqrt u + C\).

Step 6: Substitute x Back Into the Solution

In the final step, we substitute \(x^2 + 5\) back in for \(u\) to solve for \(x\) in the original function. This is \(5/\sqrt{x^2 + 5} + C \).