Question
Evaluate the following integrals:
$\int\sin^5\text{x}\cos\text{x}\text{ dx}$
$\int\sin^5\text{x}\cos\text{x}\text{ dx}$
✓
Answer
$\int\sin^5\text{x}\cos\text{x}\text{ dx}$
Let $\sin\text{x}=\text{t}$
$\cos\text{x}\text{ dx}=\text{dt}$
Now, $\int\sin^5\text{x}\cos\text{x}\text{ dx}$
$=\int\text{t}^5\text{dt}$
$=\frac{\sin^6\text{x}}{6}+\text{C}$
$=\sin\text{x}+\frac{\sin^5\text{x}}{5}-\frac{2}{3}\sin^3\text{x}$
Let $\sin\text{x}=\text{t}$
$\cos\text{x}\text{ dx}=\text{dt}$
Now, $\int\sin^5\text{x}\cos\text{x}\text{ dx}$
$=\int\text{t}^5\text{dt}$
$=\frac{\sin^6\text{x}}{6}+\text{C}$
$=\sin\text{x}+\frac{\sin^5\text{x}}{5}-\frac{2}{3}\sin^3\text{x}$
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