1.9: Indefinite Integrals of Elementary Functions

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From our known derivatives of elementary functions, we can determine some simple indefinite integrals. The power rule gives us \[\int x^n dx=\frac{x^{n+1}}{n+1}+c,\quad n\neq -1.\nonumber\]

When \(n = −1\), and \(x\) is positive, we have \[\int\frac{1}{x}dx=\ln x+c.\nonumber\]

If \(x\) is negative, using the chain rule we have \[\frac{d}{dx}\ln (-x)=\frac{1}{x}.\nonumber\]

Therefore, since \[|x|=\left\{\begin{array}{ll}-x&\text{if }x<0; \\ x&\text{if }x>0,\end{array}\right.\nonumber\] we can generalize our indefinite integral to strictly positive or strictly negative \(x\):

\[\int\frac{1}{x}dx=\ln |x|+c.\nonumber\]

Trigonometric functions can also be integrated:

\[\int\cos xdx=\sin x+c,\quad\int\sin xdx=-\cos x+c.\nonumber\]

Easily proved identities are an addition rule:

\[\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx;\nonumber\] and multiplication by a constant:

\[\int Af(x)dx=A\int f(x)dx.\nonumber\]

This permits integration of functions such as \[\int (x^2+7x+2)dx=\frac{x^3}{3}+\frac{7x^2}{2}+2x+c,\nonumber\] and \[\int (5\cos x+\sin x)dx=5\sin x-\cos x+c.\nonumber\]