Question

Va rog ajutați-mă și pe mine. Subpunctele însemnate (b, d, g, h)
Urgent

Answer 1

Vom folosi proprietatea de liniaritate a integralei si urmatoarele formule:

$\int\limits{x^a} \, dx =\frac{x^{a+1}}{a+1}+C \ , \ a\in R\char`\\ \{-1\} \\\int\limits\frac{1}{x}dx=ln|x|+C$

b)

$\int\limits(3x^2+7x+11)\,dx=3\int\limits x^2\,dx+7\int\limits x\,dx+11\int\limits x^0\,dx=\\\\=3\frac{x^3}{3}+7\frac{x^2}{2}+11\frac{x^1}{1}+C=x^3+\frac{7}{2}x^2+11x+C$

d)

$\int\limits(2x-1)^2\,dx=\int\limits(4x^2-4x+1)\,dx=4\int\limits x^2\,dx-4\int\limits x\,dx+\int\limits\,dx=\\\\=4\frac{x^3}{3}-4\frac{x^2}{2}+x+C=\frac{4}{3}x^3-2x^2+x+C$

g)

$\int\limits(\frac{2}{x^2}+\frac{1}{x^3}+\frac{1}{x^4})\,dx=2\int\limits\frac{1}{x^2}\,dx+\int\limits\frac{1}{x^3}\,dx+\int\limits\frac{1}{x^4}\,dx=\\\\=2\int\limits x^{-2}\,dx+\int\limits x^{-3}\,dx+\int\limits x^{-4}\,dx=2\frac{x^{-1}}{-1}+\frac{x^{-2}}{-2}+\frac{x^{-3}}{-3}+C=\\\\=-\frac{2}{x}-\frac{1}{2x^2}-\frac{1}{3x^3}+C$

h)

$\int\limits\frac{1+x+x^2+x^3}{x^4}\,dx=\int\limits(\frac{1}{x^4}+\frac{x}{x^4}+\frac{x^2}{x^4}+\frac{x^3}{x^4})\,dx=\int\limits(x^{-4}+x^{-3}+x^{-2}+\frac{1}{x})\,dx=\\\\=\int\limits x^{-4}\,dx+\int\limits x^{-3}\,dx+\int\limits x^{-2}\,dx+\int\limits \frac{1}{x}\,dx=-\frac{1}{3}x^{-3}-\frac{1}{2}x^{-2}-x^{-1}+ln|x|+C$