Question

Buna ma poate ajuta cineva la B ?

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Answer 1

$\displaystyle f:\mathbb{R}\rightarrow \mathbb{R},~f(x)=x^2+2\\\\ \int_0^12xf\Big(x^2\Big)dx=\int_0^12x\Big(\Big(x^2\Big)^2+2\Big)dx=\int_0^12x\Big(x^4+2\Big)dx=\\\\=\int_0^1\Big(2x^5+4x\Big)dx=\int_0^12x^5dx+\int_0^14xdx=2\int_0^1x^5dx+4\int_0^1xdx=\\\\=2\cdot \frac{x^6}{6} \Bigg|_0^1+4 \cdot \frac{x^2}{2} \Bigg|_0^1=2 \cdot \Bigg(\frac{1^6}{6} -\frac{0^6}{6} \Bigg)+4 \cdot \Bigg(\frac{1^2}{2} -\frac{0^2}{2} \Bigg)=2\cdot\frac{1}{6}+4 \cdot\frac{1}{2} =\\\\=\frac{1}{3} +2=\frac{1+6}{3} =\frac{7}{3}$