/*! 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 5 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 5

Find the indefinite integral. $$ \int \frac{x^{2}-4}{x} d x $$

Short Answer

Expert verified
\(\frac{1}{2} x^{2} - 4 \ln|x| + C\)

Step by step solution

01

Divide the Terms

Divide each term of numerator individually by the denominator term, separating it into two simpler integrals. This results in the function: \(x - \frac{4}{x}\)
02

Integrate each term

We can now take the integral of each term individually. The integral of \(x\) is \(\frac{1}{2} x^{2}\), and the integral of \(\frac{4}{x}\) is \(4 \ln|x|\). Applying these rules gives us: \(\frac{1}{2} x^{2} - 4 \ln|x|\)
03

Add Constant of Integration

The indefinite integral is ambiguous up to an arbitrary constant. Therefore, we add a constant of integration, denoted as \(+ C\), to the result. The solution thus becomes: \(\frac{1}{2} x^{2} - 4 \ln|x| + C\)

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Most popular questions from this chapter

Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \operatorname{arcsec} \frac{|u|}{a}+C $$

Verify the differentiation formula. \(\frac{d}{d x}\left[\sinh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}+1}}\)

Evaluate the integral. \(\int_{0}^{\ln 2} 2 e^{-x} \cosh x d x\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int \frac{d x}{25+x^{2}}=\frac{1}{25} \arctan \frac{x}{25}+C $$

Linear and Quadratic Approximations In Exercises 33 and 34 use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\). Use a graphing utility to graph the function and its linear and quadratic approximations. \(f(x)=\cosh x, \quad a=0\)
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