Short and Sweet Calculus

5.2 Rules for Integrating Standard Elementary Forms

In the preceding chapters, we learned a general rule for how to differentiate a function. We need to form \(\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\). However, integration has no general rule. To integrate we need to use our previous knowledge of the known results of differentiation. We have to answer this question: What function if differentiated will give the given function. This is a tentative process of finding our way back. To be able to integrate most functions in calculus, we do not need more than a few known integrals many of which we discuss in this section. The rest of the integrals can often be reduced to one of these known integrals by the following property and integration methods some of which to be discussed in this chapter.

5.2. Let \(f(x)\) and \(g(x)\) be integrable functions, and \(a\) and \(b\) be two real numbers. Then

\[\int\left[af(x)+bg(x)\right]dx=a\int f(x)dx+b\int g(x)dx.\]

  • When \(a=1\) and \(b=-1\), we have \[\int\left[f(x)-g(x)\right]dx=\int f(x)dx-\int g(x)dx.\]

  • More generally, if \(f_{1}(x),\dots,f_{n}(x)\) are integrable functions, then \[\int[a_{1}f_{1}(x)+a_{2}f_{2}(x)+\cdots+a_{n}f_{n}(x)]dx=a_{1}\int f_{1}(x)dx+a_{2}\int f_{2}(x)dx+\cdots+a_{n}\int f_{n}(x)dx,\] where \(a_{1},\dots,a_{n}\) are real numbers.

  1. \({\displaystyle \int x^{r}dx=\frac{1}{r+1}x^{r+1}+C}\qquad\) (\(r\neq-1)\)

  2. \({\displaystyle \int\frac{1}{x}dx=\ln|x|+C}\)

  3. \({\displaystyle \int\sin x}\ dx=-\cos x+C\)

  4. \({\displaystyle \int\cos x\ dx=\sin x+C}\)

  5. \({\displaystyle \int\sec^{2}x\ dx=\tan x+C}\)

  6. \({\displaystyle \int\sec x\tan x\ dx=\sec x+C}\)

  7. \({\displaystyle \int\csc^{2}x\ dx=-\cot x+C}\)

  8. \({\displaystyle \int\csc x\cot x\ dx=-\csc x+C}\)

  9. \({\displaystyle \int\frac{1}{x^{2}+1}dx=\arctan x+C}\)

  10. \({\displaystyle \int\frac{1}{\sqrt{1-x^{2}}}+C=\arcsin x+C}\)

  11. \({\displaystyle \int e^{x}dx=e^{x}+C}\)

  12. \({\displaystyle {\displaystyle \int\sinh x\ dx=\cosh x+C}}\)

  13. \({\displaystyle \int\cosh x\ dx=\sinh x+C}\)

Example 5.1. Find \({\displaystyle \int\sqrt[3]{x}dx}\).

Solution

\[\int\sqrt[3]{x}dx=\int x^{1/3}dx=\frac{1}{\frac{1}{3}+1}x^{1/3+1}+C=\frac{3}{4}x^{4/3}+C.\]

Example 5.2. Find \({\displaystyle \int\left(\frac{2a}{\sqrt{y}}-\frac{b}{y}+c\sqrt[3]{y^{2}}\right)}dy.\)

Solution

\[\begin{aligned} \int\left(\frac{2a}{\sqrt{y}}-\frac{b}{y}+c\sqrt[3]{y^{2}}\right)dy & =2a\int\frac{1}{\sqrt{y}}-b\int\frac{1}{y}dy+c\int\sqrt[3]{y^{2}}dy\\ & =2a\int y^{-1/2}dy-b\int y^{-1}dy+c\int y^{2/3}dy\\ & =2a\frac{1}{-\frac{1}{2}+1}y^{-\frac{1}{2}+1}+C_{1}+b\ln|y|+C_{2}+c\frac{1}{\frac{2}{3}+1}y^{1+\frac{2}{3}}+C_{3}\\ & =4a\sqrt{y}+b\ln|y|+\frac{3c}{5}y^{\frac{5}{3}}+C.\end{aligned}\] where \(C=C_{1}+C_{2}+C_{3}\).

  • There is no need to add a constant of integration after evaluating each integal (as is done in the last example). It is much easier to combine all of the integration constants and add only one constant of integration (denoted by \(C\) or another letter that you prefer) to the final answer.

Example 5.3. Find \({\displaystyle \int\left(\frac{x^{2}+3}{x}-\frac{2}{\sqrt{1-x^{2}}}\right)dx}\).

Solution

\[\begin{aligned} \int\left(\frac{x^{2}+3}{x}-\frac{2}{\sqrt{1-x^{2}}}\right)dx & =\int\left(x+\frac{3}{x}\right)dx-2\int\frac{1}{\sqrt{1-x^{2}}}dx\\ & =\frac{1}{2}x^{2}+3\ln|x|-2\arcsin x+C.\end{aligned}\]


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