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Q.
$\int \limits_{-\pi/2} ^{\pi /2} \sin^2x \, \cos^2x (\sin x+\cos x)dx$ is equal to
Integrals
Solution:
$\int\limits_{-\pi /2}^{\pi /2}sin^{2}\, x cos^{2}\, x \left(sin\,x+cos\,x\right)dx$
$=\int\limits_{- \pi /2}^{\pi /2}sin^{3}\,x cos^{2}\,x\, dx+\int\limits_{-\pi /2}^{\pi /2}sin^{2}\, x \, cos^{3}\,x \, dx$
[$\because$ $sin^{3}\,x cos^{2}\,x$ is odd and $sin^{2}\, x cos^{3}\,x$ is even]
$=0+2 \int \limits_{0}^{\pi/ 2} sin^{2}\, x\, cos^{3}\, x\, dx$
Put $sin x = z \therefore cosx dx - dz$
$\therefore $ given integral $=2 \int\limits_{0}^{1} z^{2}\left(1-z^{2}\right)dz$
$=2\left|\frac{z^{3}}{3}-\frac{z^{5}}{5}\right|_{0}^{1}$
$=2\left(\frac{1}{3}-\frac{1}{5}\right)=\frac{4}{15}$