Notes: What is the integral of Sin^2 x Cos^3 x dx?

Monday, March 24, 2014

What is the integral of Sin^2 x Cos^3 x dx?

$\displaystyle\int{{\sin}}^{{2}}{x}{{\cos}}^{{3}}{x}{\left.{d}{x}\right.}=\int{{\sin}}^{{2}}{x}{{\cos}}^{{2}}{x}{\cos{{x}}}$

$\displaystyle=\int{{\sin}}^{{2}}{x}{\left({1}-{{\sin}}^{{2}}{x}\right)}{\cos{{x}}}$

put

$\displaystyle\therefore$
int sin^2 x (1- sin^2 x) cos x dx = int [t^2 (1-t^2)] dt

$\displaystyle=\int{\left({{t}}^{{2}}-{{t}}^{{4}}\right)}{\left.{d}{t}\right.}$

$\displaystyle=\frac{{{{t}}^{{3}}}}{{3}}-\frac{{{{t}}^{{5}}}}{{5}}+{c}$

now plug in $\displaystyle{t}$ value as $\displaystyle{\sin{}}$
x

required integral = $\displaystyle\frac{{1}}{{3}}{{\sin}}^{{3}}{x}-\frac{{1}}{{5}}{{\sin}}^{{5}}{x}+$
c

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