Find the integrals of the functions in Exercises: ^3x ^3x - Vidyadip

Question

Find the integrals of the functions in Exercises:
$\sin^3\text{x}\cos^3\text{x}$

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Answer

$\text{Let}\text{ I}=\int\sin^3\text{x}\cos^3\text{x}\text{ dx}$
$=\int\cos^3\text{x }\sin^2\text{x }\sin\text{x}\text{ dx}$
$=\int\cos^3\text{x}\big(1-\cos^2\text{x}\big)\sin\text{x}\text{ dx}$
$\text{Let}\cos\text{x}=\text{t}$
$\Rightarrow -\sin\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\text{I}=-\int\text{t}^3\big(1-\text{t}^2\big)\text{ dt}$
$=-\int\big(\text{t}^3-\text{t}^5\big)\text{dt}$
$=-\Bigg\{\frac{\text{t}^4}{4}-\frac{\text{t}^6}{6}\Bigg\}+\text{C}$
$=-\Bigg\{\frac{\cos^4\text{x}}{4}-\frac{\cos^6\text{x}}{6}\Bigg\}+\text{C}$
$=\frac{\cos^6\text{x}}{6}-\frac{\cos^4\text{x}}{4}+\text{C}$

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