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Question Number 168296 by Florian last updated on 07/Apr/22

$$\int (5x+2)cos\left(2x\right)dx=?$$

Answered by floor(10²Eta[1]) last updated on 07/Apr/22

$$\mathrm{u}=5\mathrm{x}+2\Rightarrow \mathrm{du}=5\mathrm{dx}$$$$\mathrm{dv}=\cos \left(2\mathrm{x}\right)\mathrm{dx}\Rightarrow \mathrm{v}=\frac{\sin \left(2\mathrm{x}\right)}{2}$$$$=\frac{(5\mathrm{x}+2)\sin \left(2\mathrm{x}\right)}{2}-\frac{5}{2}\int \sin \left(2\mathrm{x}\right)\mathrm{dx}$$$$=\frac{(5\mathrm{x}+2)\sin \left(2\mathrm{x}\right)}{2}+\frac{5\cos \left(2\mathrm{x}\right)}{4}+\mathrm{C}$$$$$$

Commented by Florian last updated on 08/Apr/22

$$Correct!$$

Answered by peter frank last updated on 07/Apr/22

$$\int 5\mathrm{xcos}2\mathrm{xdx}+\int 2\cos 2\mathrm{xdx}$$$$5\int \mathrm{xcos}\mathrm{xdx}+2\int \cos 2\mathrm{xdx}$$$$\mathrm{by}\mathrm{part}$$

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