Evaluate the following integrals: ^3x dx - Vidyadip

Question

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

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Answer

Let $\text{I}=\int\text{x}\cos^3\text{x dx}$
$=\int\text{x}\Big(\frac{3\cos\text{x}+\cos3\text{x}}{4}\Big)\text{dx}$
$\frac{1}{4}\int\text{x}(3\cos\text{x}+\cos3\text{x})\text{dx}$
Using integration by parts,
$\text{I}=\frac{1}{4}\big[\text{x}\int(3\cos\text{x}+\cos3\text{x})\text{dx}-\int(1\int(3\cos\text{x}+\cos3\text{x})\text{dx})\text{dx}\big]$
$=\frac{1}{4}\Big[\text{x}\Big(3\sin\text{x}+\frac{\sin3\text{x}}{3}\Big)-\int\Big(3\sin\text{x}+\frac{\sin3\text{x}}{3}\Big)\text{dx}\Big]$
$=\frac{1}{4}\Big[3\text{x}\sin\text{x}+\frac{\text{x}\sin3\text{x}}{3}+3\cos\text{x}+\frac{\cos3\text{x}}{9}\Big]+\text{C}$
$\text{I}=\frac{3\text{x}\sin\text{x}}{4}+\frac{\text{x}\sin3\text{x}}{12}+\frac{3\cos\text{x}}{4}+\frac{\cos3\text{x}}{36}+\text{C}$

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