Question
Evaluate the following integrals:
$\int\cos^3\sqrt{\text{x}}\text{dx}$
$\int\cos^3\sqrt{\text{x}}\text{dx}$
✓
Answer
Let $\text{I}=\int\cos^3\sqrt{\text{x}}\text{dx}$
Let $\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt }$
$=2\int\text{t}\cos^3\text{t dt}$
$=2\int\text{t}\Big(\frac{3\cos\text{t}+\cos3\text{t}}{4}\Big)\text{dt}$
$=\frac{1}{2}\int\text{t}(3\cos\text{t}+\cos3\text{t})\text{dt}$
Using integral\tion by parts,
$\text{I}=\frac{1}{2}\Big[\text{t}\Big(3\sin\text{t}+\frac{1}{3}\sin3\text{t}\Big)+\int\Big(1\times3\sin\text{t}+\frac{\sin3\text{t}}{3}\Big)\text{dt}\Big]$
$=\frac{1}{2}\Big[\text{t}\Big(\frac{9\sin\text{t}+\sin3\text{t}}{3}\Big)+3\cos\text{t}+\frac{\cos3\text{t}}{9}\Big]+\text{C}$
$=\frac{1}{18}\big[27\text{t}\sin\text{t}+3\text{t}\sin3\text{t}+9\cos\text{t}+\cos3\text{t}\big]+\text{C}$
$\text{I}=\frac{1}{18}\big[27\sqrt{\text{x}}\sin\sqrt{\text{x}}+3\sqrt{\text{x}}\sin3\sqrt{\text{x}}+9\cos\sqrt{\text{x}}+\cos3\sqrt{\text{x}}\big]+\text{C}$
Let $\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt }$
$=2\int\text{t}\cos^3\text{t dt}$
$=2\int\text{t}\Big(\frac{3\cos\text{t}+\cos3\text{t}}{4}\Big)\text{dt}$
$=\frac{1}{2}\int\text{t}(3\cos\text{t}+\cos3\text{t})\text{dt}$
Using integral\tion by parts,
$\text{I}=\frac{1}{2}\Big[\text{t}\Big(3\sin\text{t}+\frac{1}{3}\sin3\text{t}\Big)+\int\Big(1\times3\sin\text{t}+\frac{\sin3\text{t}}{3}\Big)\text{dt}\Big]$
$=\frac{1}{2}\Big[\text{t}\Big(\frac{9\sin\text{t}+\sin3\text{t}}{3}\Big)+3\cos\text{t}+\frac{\cos3\text{t}}{9}\Big]+\text{C}$
$=\frac{1}{18}\big[27\text{t}\sin\text{t}+3\text{t}\sin3\text{t}+9\cos\text{t}+\cos3\text{t}\big]+\text{C}$
$\text{I}=\frac{1}{18}\big[27\sqrt{\text{x}}\sin\sqrt{\text{x}}+3\sqrt{\text{x}}\sin3\sqrt{\text{x}}+9\cos\sqrt{\text{x}}+\cos3\sqrt{\text{x}}\big]+\text{C}$
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