Evaluate the following integrals: ^4x ^3x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sin^4\text{x}\cos^3\text{x}\text{ dx}$

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Answer

$\int\sin^4\text{x}\cos^3\text{x}\text{ dx}$
$=\int\sin^4\text{x}\cdot\cos^2\text{x }\cos\text{x}\text{ dx}$
$=\int\sin^4\text{x}\big(1-\sin^2\text{x}\big)\cos\text{x}\text{ dx}$
Let $\sin\text{x}=\text{t}$
$\cos\text{x}\text{ dx}=\text{dt}$
Now, $\int\sin^4\text{x}\big(1-\sin^2\text{x}\big)\cos\text{x}\text{ dx}$
$=\int\text{t}^4(1-\text{t}^2)\text{dt}$
$=\int(\text{t}^4+\text{t}^6)=\text{dt}$
$=\frac{\text{t}^5}{5}-\frac{\text{t}^7}{7}+\text{C}$
$=\frac{\sin^5\text{x}}{5}-\frac{\sin^7\text{x}}{7}+\text{C}$

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