.Помогите решить интегралы... Created by izabella9. matematika-ru

Помогите решить интегралы...

izabella9 @izabella9

July 2022 1 22 Report

.Помогите решить интегралы..

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NNNLLL54

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$1)\; \; \int cos^2x\cdot sin^2x\, dx=\int (sinx\cdot cosx)^2\, dx=\int (\frac{1}{2}sin2x)^2\, dx=\\\\=\frac{1}{4}\int sin^22x\, dx=\frac{1}{4}\int \frac{1-cos4x}{2}\, dx=\frac{1}{8}\int (1-cos4x)\, dx=\\\\=\frac{1}{8}\cdot (x-\frac{1}{4}sin4x)+C\; ;\\\\\\2)\; \; \int cos^3x\, dx=\int cos^2x\cdot \underbrace {cosx\, dx}_{d(sinx)}=\int (1-sin^2x)\, d(sinx)=\\\\=\int d(sinx)-\int sin^2x\cdot d(sinx)=sinx-\frac{sin^3x}{3}+C\; .\\\\\\\star \; \; \int cos(kx+b)\, dx=\frac{1}{k}\cdot sin(kx+b)+C\; \; \star$

$\star \; \; d(sinx)=(sinx)'\cdot dx=cosx\cdot dx\; \; \star$

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