0.9: Indefinite Integrals of Elementary Functions
From our known derivatives of elementary functions, we can determine some simple indefinite integrals. The power rule gives us
\[\int x^ndx=\frac{x^{n+1}}{n+1}+c,\quad n\neq -1.\nonumber \]
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
\[\int (5\cos x+\sin x)dx=5\sin x-\cos x+c.\nonumber \]