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I have a problem with this integral: $\int e^{3t}(\cos(t))^{3}dt$

I try to do this by parts or substitution but I did not manage to do count this. I will be glad for any tips to solve this the fastest way.

HeroZhang001
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Marcin Majewski
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  • or use the fact $\cos(t)=\frac{\mathrm{e}^{-it} + \mathrm{e}^{it}}{2}$. Then expand the equation and collect terms and integrate with respect t. – Chinny84 Jan 24 '14 at 18:22

2 Answers2

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HINT:

Use $$\cos3t=4\cos^3t-3\cos t\iff \cos t=\frac{\cos3t+3\cos t}4$$

then this or utilize Integrate $e^{ax}\sin(bx)?$

Community
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lab bhattacharjee
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use the trigonometric identity $$\cos 3x=4\cos^3x-3\cos x$$ the integral becomes $$\int e^{3t}\frac{1}{4}(\cos 3x +3\cos x)dx$$ now apply integration by parts.

Suraj M S
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