/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 23

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

Short Answer

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

Step by step solution

01

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.
02

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 \(du = 2x dx\).
03

Rearrange the Equation for dx

Rearrange the equation from step 2 to solve for \(dx\). This gives \(dx=du/(2x)\).
04

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 \(\int (鈭�5 / 2) (1/u^{3/2}) du\).
05

Solve the Integral

Now solve this new, simpler integral. That is, the integral is \(-(5/2) \int (1/u^{3/2}) du = -(5/2)(-2/(\sqrt u)) + C = 5/\sqrt u + C\).
06

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 \(5/\sqrt{x^2 + 5} + C \).

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