`int (cos(3 theta) - 1) d theta` Find the indefinite integral.

Wednesday, September 7, 2011

$\displaystyle\int{\left({\cos{{\left({3}\theta\right)}}}-{1}\right)}{d}\theta$ Find the indefinite integral.

$\displaystyle\int{\left({\cos{{\left({3}\theta\right)}}}-{1}\right)}{d}\theta=$

Use additivity of integral: $\displaystyle\int{\left({f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}=\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}+\int{g{{\left({x}\right)}}}{\left.{d}{x}\right.}.$ $\displaystyle\int{\cos{{\left({3}\theta\right)}}}{d}\theta-\int{d}\theta=$

Since the second integral is easy $\displaystyle\int{d}\theta=\theta+{C}$ we will concentrate on the first integral. To solve the first integral we will make substitution $\displaystyle{u}={3}\theta,$ $\displaystyle{d}{u}={3}{d}\theta\Rightarrow{d}\theta=\frac{{{d}{u}}}{{3}}$